# 1.3: Basic Classes of Functions - Mathematics

We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

## Linear Functions and Slope

The easiest type of function to consider is a linear function. Linear functions have the form (f(x)=ax+b), where (a) and (b) are constants. In Figure (PageIndex{1}), we see examples of linear functions when a is positive, negative, and zero. Note that if (a>0), the graph of the line rises as (x) increases. In other words, (f(x)=ax+b) is increasing on ((−∞, ∞)). If (a<0), the graph of the line falls as (x) increases. In this case, (f(x)=ax+b) is decreasing on ((−∞, ∞)). If (a=0), the line is horizontal.

Figure (PageIndex{1}): These linear functions are increasing or decreasing on ((∞, ∞)) and one function is a horizontal line.

As suggested by Figure, the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in (y) for each unit change in (x). The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in (y) versus the change in (x). To do so, we choose any two points ((x_1,y_1)) and ((x_2,y_2)) on the line and calculate (dfrac{y_2−y_1}{x_2−x_1}). In Figure (PageIndex{2}), we see this ratio is independent of the points chosen.

Figure (PageIndex{2}): For any linear function, the slope ((y_2−y_1)/(x_2−x_1)) is independent of the choice of points ((x_1,y_1)) and ((x_2,y_2)) on the line.

Definition: Linear Functions

Consider line (L) passing through points ((x_1,y_1)) and ((x_2,y_2)). Let (Δy=y_2−y_1) and (Δx=x_2−x_1) denote the changes in (y) and (x),respectively. The slope of the line is

[m=dfrac{y_2−y_1}{x_2−x_1}=dfrac{Δy}{Δx}]

We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula (f(x)=ax+b). As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating ((y_2−y_1)/(x_2−x_1)) for any points ((x_1,y_1)) and ((x_2,y_2)) on the line. Evaluating the function (f) at (x=0), we see that ((0,b)) is a point on this line. Evaluating this function at (x=1), we see that ((1,a+b)) is also a point on this line. Therefore, the slope of this line is

[dfrac{(a+b)−b}{1−0}=a.]

We have shown that the coefficient (a) is the slope of the line. We can conclude that the formula (f(x)=ax+b) describes a line with slope (a). Furthermore, because this line intersects the (y)-axis at the point ((0,b)), we see that the y-intercept for this linear function is ((0,b)). We conclude that the formula (f(x)=ax+b) tells us the slope, a, and the (y)-intercept, ((0,b)), for this line. Since we often use the symbol (m) to denote the slope of a line, we can write

[f(x)=mx+b]

to denote the slope-intercept form of a linear function.

Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point ((x_1,y_1)) and the slope of the line is (m). Since any other point ((x,f(x))) on the graph of (f) must satisfy the equation

[m=dfrac{f(x)−y_1}{x−x_1},]

this linear function can be expressed by writing

[f(x)−y_1=m(x−x_1).]

We call this equation the point-slope equation for that linear function.

Since every nonvertical line is the graph of a linear function, the points on a nonvertical line can be described using the slope-intercept or point-slope equations. However, a vertical line does not represent the graph of a function and cannot be expressed in either of these forms. Instead, a vertical line is described by the equation (x=k) for some constant (k). Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation

[ax+by=c,]

where (a,b) are both not zero, to denote the standard form of a line.

Definition: point-slope equation, point-slope equation and the standard form of a line

Consider a line passing through the point ((x_1,y_1)) with slope (m). The equation

[y−y_1=m(x−x_1)]

is the point-slope equation for that line.

Consider a line with slope (m) and (y)-intercept ((0,b).) The equation

[y=mx+b]

is an equation for that line in point-slope equation.

The standard form of a line is given by the equation

[ax+by=c,]

where (a) and (b) are both not zero. This form is more general because it allows for a vertical line, (x=k).

Example (PageIndex{1}): Finding the Slope and Equations of Lines

Consider the line passing through the points ((11,−4)) and ((−4,5)), as shown in Figure.

Figure (PageIndex{3}): Finding the equation of a linear function with a graph that is a line between two given points.

1. Find the slope of the line.
2. Find an equation for this linear function in point-slope form.
3. Find an equation for this linear function in slope-intercept form.

Solution

1. The slope of the line is

[m=dfrac{y_2−y_1}{x_2−x_1}=dfrac{5−(−4)}{−4−11}=−dfrac{9}{15}=−dfrac{3}{5}.]

2. To find an equation for the linear function in point-slope form, use the slope (m=−3/5) and choose any point on the line. If we choose the point ((11,−4)), we get the equation

[f(x)+4=−dfrac{3}{5}(x−11).]

3. To find an equation for the linear function in slope-intercept form, solve the equation in part b. for (f(x)). When we do this, we get the equation

[f(x)=−dfrac{3}{5}x+dfrac{13}{5}.]

Exercise (PageIndex{1})

Consider the line passing through points ((−3,2)) and ((1,4)).

1. Find the slope of the line.
2. Find an equation of that line in point-slope form.
3. Find an equation of that line in slope-intercept form.
Hint

The slope (m=Δy/Δx).

(m=1/2).

The point-slope form is (y−4=dfrac{1}{2}(x−1)).

The slope-intercept form is (y=dfrac{1}{2}x+dfrac{7}{2}).

Example (PageIndex{2}):

Jessica leaves her house at 5:50 a.m. and goes for a 9-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Jessica runs at a constant pace.

1. Describe the distance (D) (in miles) Jessica runs as a linear function of her run time (t) (in minutes).
2. Sketch a graph of (D).
3. Interpret the meaning of the slope.

Solution:

a. At time (t=0), Jessica is at her house, so (D(0)=0). At time (t=78) minutes, Jessica has finished running (9) mi, so (D(78)=9). The slope of the linear function is

(m=dfrac{9−0}{78−0}=dfrac{3}{26}.)

The (y)-intercept is ((0,0)), so the equation for this linear function is

(D(t)=dfrac{3}{26}t.)

b. To graph (D), use the fact that the graph passes through the origin and has slope (m=3/26.)

c. The slope (m=3/26≈0.115) describes the distance (in miles) Jessica runs per minute, or her average velocity.

## Polynomials

A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form

[f(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0]

for some integer (n≥0) and constants (a_n,a+{n−1},…,a_0), where (a_n≠0). In the case when (n=0), we allow for (a_0=0); if (a_0=0), the function (f(x)=0) is called the zero function. The value (n) is called the degree of the polynomial; the constant an is called the leading coefficient. A linear function of the form (f(x)=mx+b) is a polynomial of degree 1 if (m≠0) and degree 0 if (m=0). A polynomial of degree 0 is also called a constant function. A polynomial function of degree 2 is called a quadratic function. In particular, a quadratic function has the form (f(x)=ax^2+bx+c), where (a≠0). A polynomial function of degree (3) is called a cubic function.

## Power Functions

Some polynomial functions are power functions. A power function is any function of the form (f(x)=ax^b), where (a) and (b) are any real numbers. The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. (We consider other cases later.) If the exponent is a positive integer, then (f(x)=ax^n) is a polynomial. If (n) is even, then (f(x)=ax^n) is an even function because (f(−x)=a(−x)^n=ax^n) if (n) is even. If (n) is odd, then (f(x)=ax^n) is an odd function because (f(−x)=a(−x)^n=−ax^n) if (n) is odd (Figure (PageIndex{3})).

Figure (PageIndex{4}): (a) For any even integer (n),(f(x)=ax^n) is an even function. (b) For any odd integer (n),(f(x)=ax^n) is an odd function.

## Behavior at Infinity

To determine the behavior of a function (f) as the inputs approach infinity, we look at the values (f(x)) as the inputs, (x), become larger. For some functions, the values of (f(x)) approach a finite number. For example, for the function (f(x)=2+1/x), the values (1/x) become closer and closer to zero for all values of (x) as they get larger and larger. For this function, we say “(f(x)) approaches two as x goes to infinity,” and we write f(x)→2 as x→∞. The line y=2 is a horizontal asymptote for the function (f(x)=2+1/x) because the graph of the function gets closer to the line as (x) gets larger.

For other functions, the values (f(x)) may not approach a finite number but instead may become larger for all values of x as they get larger. In that case, we say “(f(x)) approaches infinity as (x) approaches infinity,” and we write (f(x)→∞) as (x→∞). For example, for the function (f(x)=3x^2), the outputs (f(x)) become larger as the inputs (x) get larger. We can conclude that the function (f(x)=3x^2) approaches infinity as (x) approaches infinity, and we write (3x^2→∞) as (x→∞). The behavior as (x→−∞) and the meaning of (f(x)→−∞) as (x→∞) or (x→−∞) can be defined similarly. We can describe what happens to the values of (f(x)) as (x→∞) and as (x→−∞) as the end behavior of the function.

To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function (f(x)=ax^2+bx+c). If (a>0), the values (f(x)→∞) as (x→±∞). If (a<0), the values (f(x)→−∞) as (x→±∞). Since the graph of a quadratic function is a parabola, the parabola opens upward if (a>0).; the parabola opens downward if (a<0) (Figure (PageIndex{4a})).

Now consider a cubic function (f(x)=ax^3+bx^2+cx+d). If (a>0), then (f(x)→∞) as (x→∞) and (f(x)→−∞) as (x→−∞). If (a<0), then (f(x)→−∞) as (x→∞) and (f(x)→∞) as (x→−∞). As we can see from both of these graphs, the leading term of the polynomial determines the end behavior (Figure (PageIndex{4b})).

Figure (PageIndex{5}): (a) For a quadratic function, if the leading coefficient (a>0),the parabola opens upward. If (a<0), the parabola opens downward. (b) For a cubic function (f), if the leading coefficient (a>0), the values (f(x)→∞) as (x→∞) and the values (f(x)→−∞) as (x→−∞). If the leading coefficient (a<0), the opposite is true.

## Zeros of Polynomial Functions

Another characteristic of the graph of a polynomial function is where it intersects the (x)-axis. To determine where a function f intersects the (x)-axis, we need to solve the equation (f(x)=0) for (n) the case of the linear function (f(x)=mx+b), the (x)-intercept is given by solving the equation (mx+b=0). In this case, we see that the (x)-intercept is given by ((−b/m,0)). In the case of a quadratic function, finding the (x)-intercept(s) requires finding the zeros of a quadratic equation: (ax^2+bx+c=0). In some cases, it is easy to factor the polynomial (ax^2+bx+c) to find the zeros. If not, we make use of the quadratic formula.

[ax^2+bx+c=0,]

where (a≠0). The solutions of this equation are given by the quadratic formula

If the discriminant (b^2−4ac>0), Equation ef{quad} tells us there are two real numbers that satisfy the quadratic equation. If (b^2−4ac=0), this formula tells us there is only one solution, and it is a real number. If (b^2−4ac<0), no real numbers satisfy the quadratic equation.

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the x-axis. In some instances, it is possible to find the (x)-intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the (x)-intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the (x)-intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.

Example (PageIndex{3}): Graphing Polynomial Functions

For the following functions,

1. (f(x)=−2x^2+4x−1)
2. (f(x)=x^3−3x^2−4x)
1. describe the behavior of (f(x)) as (x→±∞),
2. find all zeros of (f), and
3. sketch a graph of (f).

Solution

1.The function (f(x)=−2x^2+4x−1) is a quadratic function.

1.Because (a=−2<0),as (x→±∞,f(x)→−∞.)

2. To find the zeros of (f), use the quadratic formula. The zeros are

(x=−4±dfrac{sqrt{4^2−4(−2)(−1)}}{2(−2)}=dfrac{−4±sqrt{8}}{−4}=dfrac{−4±2sqrt{2}}{−4}=dfrac{2±2sqrt{2}}{2}.)

3.To sketch the graph of (f),use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward.

2. The function (f(x)=x^3−3x^2−4x) is a cubic function.

1.Because (a=1>0),as (x→∞), (f(x)→∞). As (x→−∞), (f(x)→−∞).

2.To find the zeros of (f), we need to factor the polynomial. First, when we factor (x|) out of all the terms, we find

(f(x)=x(x^2−3x−4).)

Then, when we factor the quadratic function (x^2−3x−4), we find

(f(x)=x(x−4)(x+1).)

Therefore, the zeros of f are (x=0,4,−1).

3. Combining the results from parts i. and ii., draw a rough sketch of (f).

Exercise (PageIndex{2})

Consider the quadratic function (f(x)=3x^2−6x+2.) Find the zeros of (f). Does the parabola open upward or downward?

Hint

The zeros are (x=1±sqrt{3}/3). The parabola opens upward.

## Mathematical Models

A large variety of real-world situations can be described using mathematical models. A mathematical model is a method of simulating real-life situations with mathematical equations. Physicists, engineers, economists, and other researchers develop models by combining observation with quantitative data to develop equations, functions, graphs, and other mathematical tools to describe the behavior of various systems accurately. Models are useful because they help predict future outcomes. Examples of mathematical models include the study of population dynamics, investigations of weather patterns, and predictions of product sales.

As an example, let’s consider a mathematical model that a company could use to describe its revenue for the sale of a particular item. The amount of revenue (R) a company receives for the sale of n items sold at a price of (p) dollars per item is described by the equation (R=p⋅n). The company is interested in how the sales change as the price of the item changes. Suppose the data in Table show the number of units a company sells as a function of the price per item.

 (p) (n) 6 8 10 12 14 19.4 18.5 16.2 13.8 12.2

In Figure, we see the graph the number of units sold (in thousands) as a function of price (in dollars). We note from the shape of the graph that the number of units sold is likely a linear function of price per item, and the data can be closely approximated by the linear function (n= −1.04p+26) for (0≤p≤25), where (n) predicts the number of units sold in thousands. Using this linear function, the revenue (in thousands of dollars) can be estimated by the quadratic function

[R(p)=p⋅ (−1.04p+26)=−1.04p^2+26p]

for (0≤p≤25) In Example (PageIndex{1}), we use this quadratic function to predict the amount of revenue the company receives depending on the price the company charges per item. Note that we cannot conclude definitively the actual number of units sold for values of (p), for which no data are collected. However, given the other data values and the graph shown, it seems reasonable that the number of units sold (in thousands) if the price charged is (p) dollars may be close to the values predicted by the linear function (n=−1.04p+26.)

Figure (PageIndex{6}): The data collected for the number of items sold as a function of price is roughly linear. We use the linear function (n=−1.04p+26) to estimate this function.

Example (PageIndex{4}): Maximizing Revenue

A company is interested in predicting the amount of revenue it will receive depending on the price it charges for a particular item. Using the data from Table, the company arrives at the following quadratic function to model revenue (R) as a function of price per item (p:)

[R(p)=p⋅(−1.04p+26)=−1.04p^2+26p]

for 0≤p≤25.

1. Predict the revenue if the company sells the item at a price of (p=$5) and (p=$17).
2. Find the zeros of this function and interpret the meaning of the zeros.
3. Sketch a graph of (R).
4. Use the graph to determine the value of (p) that maximizes revenue. Find the maximum revenue.

Solution

a. Evaluating the revenue function at (p=5) and (p=17), we can conclude that

(R(5)=−1.04(5)^2+26(5)=104,so revenue=$104,000;) (R(17)=−1.04(17)^2+26(17)=141.44,so revenue=$144,440.)

b. The zeros of this function can be found by solving the equation (−1.04p^2+26p=0). When we factor the quadratic expression, we get (p(−1.04p+26)=0). The solutions to this equation are given by (p=0,25). For these values of (p), the revenue is zero. When (p=$0), the revenue is zero because the company is giving away its merchandise for free. When (p=$25),the revenue is zero because the price is too high, and no one will buy any items.

c. Knowing the fact that the function is quadratic, we also know the graph is a parabola. Since the leading coefficient is negative, the parabola opens downward. One property of parabolas is that they are symmetric about the axis, so since the zeros are at (p=0) and (p=25), the parabola must be symmetric about the line halfway between them, or (p=12.5).

d. The function is a parabola with zeros at (p=0) and (p=25), and it is symmetric about the line (p=12.5), so the maximum revenue occurs at a price of (p=$12.50) per item. At that price, the revenue is (R(p)=−1.04(12.5)^2+26(12.5)=$162,500.)

## Algebraic Functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form (f(x)=p(x)/q(x)),where (p(x)) and (q(x)) are polynomials. For example,

(f(x)=dfrac{3x−1}{5x+2}) and (g(x)=dfrac{4}{x^2+1})

are rational functions. A root function is a power function of the form (f(x)=x^{1/n}), where n is a positive integer greater than one. For example, f(x)=x1/2=x√ is the square-root function and (g(x)=x^{1/3}=sqrt[3]{x})) is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, (f(x)=sqrt{4−x^2}) is an algebraic function.

Example (PageIndex{5}): Finding Domain and Range for Algebraic Functions

For each of the following functions, find the domain and range.

1. (f(x)=dfrac{3x−1}{5x+2})
2. (f(x)=sqrt{4−x^2})

Solution

1.It is not possible to divide by zero, so the domain is the set of real numbers (x) such that (x≠−2/5). To find the range, we need to find the values (y) for which there exists a real number (x) such that

(y=dfrac{3x−1}{5x+2})

When we multiply both sides of this equation by (5x+2), we see that (x) must satisfy the equation

(5xy+2y=3x−1.)

From this equation, we can see that (x) must satisfy

(2y+1=x(3−5y).)

If y=(3/5), this equation has no solution. On the other hand, as long as (y≠3/5),

(x=dfrac{2y+1}{3−5y})

satisfies this equation. We can conclude that the range of (f) is ({y|y≠3/5}).

2. To find the domain of (f), we need (4−x^2≥0). When we factor, we write (4−x^2=(2−x)(2+x)≥0). This inequality holds if and only if both terms are positive or both terms are negative. For both terms to be positive, we need to find (x) such that

(2−x≥0) and (2+x≥0.)

These two inequalities reduce to (2≥x) and (x≥−2). Therefore, the set ({x|−2≤x≤2}) must be part of the domain. For both terms to be negative, we need

(2−x≤0) and (2+x≥0.)

These two inequalities also reduce to (2≤x) and (x≥−2). There are no values of (x) that satisfy both of these inequalities. Thus, we can conclude the domain of this function is ({x|−2≤x≤2}.)

If (−2≤x≤2), then (0≤4−x^2≤4). Therefore, (0≤sqrt{4−x2}≤2), and the range of (f) is ({y|0≤y≤2}.)

Exercise (PageIndex{3})

Find the domain and range for the function (f(x)=(5x+2)/(2x−1).)

Hint

The denominator cannot be zero. Solve the equation (y=(5x+2)/(2x−1)) for (x) to find the range.

The domain is the set of real numbers (x) such that (x≠1/2). The range is the set ({y|y≠5/2}).

The root functions (f(x)=x^{1/n}) have defining characteristics depending on whether (n) is odd or even. For all even integers (n≥2), the domain of (f(x)=x^{1/n}) is the interval ([0,∞)). For all odd integers (n≥1), the domain of (f(x)=x^{1/n}) is the set of all real numbers. Since (x^{1/n}=(−x)^{1/n}) for odd integers (n) ,(f(x)=x^{1/n}) is an odd function if(n) is odd. See the graphs of root functions for different values of (n) in Figure.

Figure (PageIndex{7}): (a) If (n) is even, the domain of (f(x)=sqrt[n]{x}) is ([0,∞)). (b) If (n) is odd, the domain of (f(x)=dfrac[n]{x}) is ((−∞,∞)) and the function (f(x)=dfrac[n]{x}) is an odd function.

Example (PageIndex{6}): Finding Domains for Algebraic Functions

For each of the following functions, determine the domain of the function.

1. (f(x)=dfrac{3}{x^2−1})
2. (f(x)=dfrac{2x+5}{3x^2+4})
3. (f(x)=sqrt{4−3x})
4. (f(x)=sqrt[3]{2x−1})

Solution

1. You cannot divide by zero, so the domain is the set of values (x) such that (x^2−1≠0). Therefore, the domain is ({x|x≠±1}).
2. You need to determine the values of (x) for which the denominator is zero. Since (3x^2+4≥4) for all real numbers (x), the denominator is never zero. Therefore, the domain is ((−∞,∞).)
3. Since the square root of a negative number is not a real number, the domain is the set of values (x) for which (4−3x≥0). Therefore, the domain is ({x|x≤4/3}.)
4. The cube root is defined for all real numbers, so the domain is the interval ((−∞, ∞).)

Exercise (PageIndex{4})

Find the domain for each of the following functions: (f(x)=(5−2x)/(x^2+2)) and (g(x)=sqrt{5x−1}).

Hint

Determine the values of (x) when the expression in the denominator of (f) is nonzero, and find the values of (x) when the expression inside the radical of (g) is nonnegative.

The domain of (f) is ((−∞, ∞)). The domain of (g) is ({x|x≥1/5}.)

## Transcendental Functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are (sinx, cosx, tanx, cotx, secx, and cscx.) (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form (f(x)=b^x), where the base (b>0,b≠1). A logarithmic function is a function of the form (f(x)=log_b(x)) for some constant (b>0,b≠1,) where (log_b(x)=y) if and only if (b^y=x). (We also discuss exponential and logarithmic functions later in the chapter.)

Example (PageIndex{7}): Classifying Algebraic and Transcendental Functions

Classify each of the following functions, a. through c., as algebraic or transcendental.

1. (f(x)=dfrac{sqrt{x^3+1}}{4x+2})
2. (f(x)=2^{x^2})
3. ( f(x)=sin(2x))

Solution

1. Since this function involves basic algebraic operations only, it is an algebraic function.
2. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
3. As in part b, this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.

Exercise (PageIndex{5}):

Is (f(x)=x/2) an algebraic or a transcendental function?

Algebraic

## Piecewise-Defined Functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function. The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of (x):

[f(x)=egin{cases}−x & x<0x & x≥0end{cases}.]

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for (xa), we need to pay special attention to what happens at (x=a) when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at (x=a). We examine this in the next example.

Example (PageIndex{8}): Graphing a Piecewise-Defined Function

Sketch a graph of the following piecewise-defined function:

[f(x)=egin{cases}x+3,&x<1(x−2)^2,&x≥1end{cases}.]

Solution

Graph the linear function (y=x+3) on the interval ((−∞,1)) and graph the quadratic function (y=(x−2)^2) on the interval ([1,∞)). Since the value of the function at (x=1) is given by the formula (f(x)=(x−2)^2), we see that (f(1)=1). To indicate this on the graph, we draw a closed circle at the point ((1,1)). The value of the function is given by (f(x)=x+2) for all (x<1), but not at (x=1). To indicate this on the graph, we draw an open circle at ((1,4)).

Figure (PageIndex{8}): This piecewise-defined function is linear for (x<1) and quadratic for (x≥1.)

2) Sketch a graph of the function

(f(x)=egin{cases}2−x,&x≤2x+2,&x>2end{cases}.)

Solution:

Example (PageIndex{9}): Parking Fees Described by a Piecewise-Defined Function

In a big city, drivers are charged variable rates for parking in a parking garage. They are charged $10 for the first hour or any part of the first hour and an additional$2 for each hour or part thereof up to a maximum of \$30 for the day. The parking garage is open from 6 a.m. to 12 midnight.

1. Write a piecewise-defined function that describes the cost (C) to park in the parking garage as a function of hours parked (x).
2. Sketch a graph of this function (C(x).)

Solution

1.Since the parking garage is open 18 hours each day, the domain for this function is ({x|0

[C(x)=egin{cases}10,&0

2.The graph of the function consists of several horizontal line segments.

Exercise (PageIndex{6})

The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is (49¢) for the first ounce and (21¢) for each additional ounce. Write a piecewise-defined function describing the cost (C) as a function of the weight (x) for (0

Hint

The piecewise-defined function is constant on the intervals (0,1],(1,2],….

[C(x)=egin{cases}49,&0

## Transformations of Functions

We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function (y=x^2) to get the function( f(x)=(x−2)^2). This subtraction represents a shift of the function (y=x^2) two units to the right. A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes.

A vertical shift of a function occurs if we add or subtract the same constant to each output (y). For (c>0), the graph of (f(x)+c) is a shift of the graph of (f(x)) up c units, whereas the graph of (f(x)−c) is a shift of the graph of (f(x)) down (c) units. For example, the graph of the function (f(x)=x^3+4) is the graph of (y=x^3) shifted up (4) units; the graph of the function (f(x)=x^3−4) is the graph of (y=x^3) shifted down (4) units (Figure (PageIndex{6})).

Figure (PageIndex{9}): (a) For (c>0), the graph of (y=f(x)+c) is a vertical shift up (c) units of the graph of (y=f(x)). (b) For (c>0), the graph of (y=f(x)−c) is a vertical shift down c units of the graph of (y=f(x)).

A horizontal shift of a function occurs if we add or subtract the same constant to each input (x). For (c>0), the graph of (f(x+c)) is a shift of the graph of (f(x)) to the left (c) units; the graph of (f(x−c)) is a shift of the graph of (f(x)) to the right (c) units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example.

Consider the function (f(x)=|x+3|) and evaluate this function at (x−3). Since (f(x−3)=|x|) and (x−3

Figure (PageIndex{10}): (a) For (c>0), the graph of (y=f(x+c)) is a horizontal shift left (c) units of the graph of (y=f(x)). (b) For (c>0), the graph of (y=f(x−c)) is a horizontal shift right (c) units of the graph of (y=f(x).)

A vertical scaling of a graph occurs if we multiply all outputs (y) of a function by the same positive constant. For (c>0), the graph of the function (cf(x)) is the graph of (f(x)) scaled vertically by a factor of (c). If (c>1), the values of the outputs for the function (cf(x)) are larger than the values of the outputs for the function (f(x)); therefore, the graph has been stretched vertically. If (0

Figure (PageIndex{11}): (a) If (c>1), the graph of (y=cf(x)) is a vertical stretch of the graph of (y=f(x)). (b) If (0

The horizontal scaling of a function occurs if we multiply the inputs (x) by the same positive constant. For (c>0), the graph of the function (f(cx)) is the graph of (f(x)) scaled horizontally by a factor of (c). If (c>1), the graph of (f(cx)) is the graph of (f(x)) compressed horizontally. If (0

Figure (PageIndex{12}): (a) If (c>1), the graph of (y=f(cx)) is a horizontal compression of the graph of (y=f(x)). (b) If (0

We have explored what happens to the graph of a function (f) when we multiply (f) by a constant (c>0) to get a new function (cf(x)). We have also discussed what happens to the graph of a function (f)when we multiply the independent variable (x) by (c>0) to get a new function (f(cx)). However, we have not addressed what happens to the graph of the function if the constant (c) is negative. If we have a constant (c<0), we can write (c) as a positive number multiplied by (−1); but, what kind of transformation do we get when we multiply the function or its argument by (−1?) When we multiply all the outputs by (−1), we get a reflection about the (x)-axis. When we multiply all inputs by (−1), we get a reflection about the (y)-axis. For example, the graph of (f(x)=−(x^3+1)) is the graph of (y=(x^3+1)) reflected about the (x)-axis. The graph of (f(x)=(−x)^3+1) is the graph of (y=x^3+1) reflected about the (y)-axis (Figure (PageIndex{10})).

Figure (PageIndex{13}): (a) The graph of (y=−f(x)) is the graph of (y=f(x)) reflected about the (x)-axis. (b) The graph of (y=f(−x)) is the graph of (y=f(x)) reflected about the (y)-axis.

If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function (f(x)), the graph of the related function (y=cf(a(x+b))+d) can be obtained from the graph of (y=f(x))by performing the transformations in the following order.

• Horizontal shift of the graph of (y=f(x)). If (b>0), shift left. If (b<0) shift right.
• Horizontal scaling of the graph of (y=f(x+b)) by a factor of (|a|). If (a<0), reflect the graph about the (y)-axis.
• Vertical scaling of the graph of (y=f(a(x+b))) by a factor of (|c|). If (c<0), reflect the graph about the (x) -axis.
• Vertical shift of the graph of (y=cf(a(x+b))). If (d>0), shift up. If (d<0), shift down.

We can summarize the different transformations and their related effects on the graph of a function in the following table.

Transformation of (f (c>0))Effect of the graph of (f)
(f(x)+c)Vertical shift up (c) units
(f(x)-c)Vertical shift down (c) units
(f(x+c))Shift left by (c) units
(f(x-c))Shift right by (c) units
(cf(x))

Vertical stretch if (c>1);

vertical compression if (0

(f(cx))

Horizontal stretch if (0

horizontal compression if (c>1)

Example (PageIndex{10}): Transforming a Function

For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function.

1. (f(x)=−|x+2|−3)
2. (f(x)=sqrt[3]{x}+1)

Solution:

1.Starting with the graph of (y=|x|), shift (2) units to the left, reflect about the (x)-axis, and then shift down (3) units.

Figure (PageIndex{14}): The function (f(x)=−|x+2|−3) can be viewed as a sequence of three transformations of the function (y=|x|).

2. Starting with the graph of y=x√, reflect about the y-axis, stretch the graph vertically by a factor of 3, and move up 1 unit.

Figure (PageIndex{15}): The function (f(x)=sqrt[3]{x}+1)can be viewed as a sequence of three transformations of the function (y=sqrt{x}).

Exercise (PageIndex{7})

Describe how the function (f(x)=−(x+1)^2−4) can be graphed using the graph of (y=x^2) and a sequence of transformations

Shift the graph (y=x^2) to the left 1 unit, reflect about the (x)-axis, then shift down 4 units.

### Key Concepts

• The power function (f(x)=x^n) is an even function if n is even and (n≠0), and it is an odd function if (n) is odd.
• The root function (f(x)=x^{1/n}) has the domain ([0,∞)) if n is even and the domain ((−∞,∞)) if (n) is odd. If (n) is odd, then (f(x)=x^{1/n}) is an odd function.
• The domain of the rational function (f(x)=p(x)/q(x)), where (p(x)) and (q(x)) are polynomial functions, is the set of x such that (q(x)≠0).
• Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
• A polynomial function (f) with degree (n≥1) satisfies (f(x)→±∞) as (x→±∞). The sign of the output as (x→∞) depends on the sign of the leading coefficient only and on whether (n) is even or odd.
• Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the (x)- and (y)-axes are examples of transformations of functions.

## Key Equations

• Point-slope equation of a line

(y−y1=m(x−x_1))

• Slope-intercept form of a line

(y=mx+b)

• Standard form of a line

(ax+by=c)

• Polynomial function

(f(x)=a_n^{x^n}+a_{n−1}x^{n−1}+⋯+a_1x+a_0)

## Glossary

algebraic function
a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable (x)
cubic function
a polynomial of degree 3; that is, a function of the form (f(x)=ax^3+bx^2+cx+d), where (a≠0)
degree
for a polynomial function, the value of the largest exponent of any term
linear function
a function that can be written in the form (f(x)=mx+b)
logarithmic function
a function of the form (f(x)=log_b(x)) for some base (b>0,b≠1) such that (y=log_b(x)) if and only if (b^y=x)
mathematical model
A method of simulating real-life situations with mathematical equations
piecewise-defined function
a function that is defined differently on different parts of its domain
point-slope equation
equation of a linear function indicating its slope and a point on the graph of the function
polynomial function
a function of the form (f(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0)
power function
a function of the form (f(x)=x^n) for any positive integer (n≥1)
a polynomial of degree 2; that is, a function of the form (f(x)=ax^2+bx+c) where (a≠0)
rational function
a function of the form (f(x)=p(x)/q(x)), where (p(x)) and (q(x)) are polynomials
root function
a function of the form (f(x)=x^{1/n}) for any integer (n≥2)
slope
the change in y for each unit change in x
slope-intercept form
equation of a linear function indicating its slope and y-intercept
transcendental function
a function that cannot be expressed by a combination of basic arithmetic operations
transformation of a function
a shift, scaling, or reflection of a function

## One-to-One Functions

A function f is 1 -to- 1 if no two elements in the domain of f correspond to the same element in the range of f . In other words, each x in the domain has exactly one image in the range. And, no y in the range is the image of more than one x in the domain.

If the graph of a function f is known, it is easy to determine if the function is 1 -to- 1 . Use the Horizontal Line Test. If no horizontal line intersects the graph of the function f in more than one point, then the function is 1 -to- 1 .

A function f has an inverse f &minus 1 (read f inverse) if and only if the function is 1 -to- 1 .

## Cubic Functions

A cubic function is one in the form f ( x ) = a x 3 + b x 2 + c x + d .

The "basic" cubic function, f ( x ) = x 3 , is graphed below.

The function of the coefficient a in the general equation is to make the graph "wider" or "skinnier", or to reflect it (if negative):

The constant d in the equation is the y -intercept of the graph.

The effects of b and c on the graph are more complicated. However, if you can factor the right side of the equation, you can find one or more x -intercepts , and use these to sketch the graph. (Some cubics, however, cannot be factored.)

A cubic function may have one, two or three x -intercepts, corresponding to the real roots of the related cubic equation.

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## 1.3: Basic Classes of Functions - Mathematics

A Boolean function is described by an algebraic expression consisting of binary variables, the constants 0 and 1, and the logic operation symbols
For a given set of values of the binary variables involved, the boolean function can have a value of 0 or 1. For example, the boolean function is defined in terms of three binary variables . The function is equal to 1 if and simultaneously or .
Every boolean function can be expressed by an algebraic expression, such as one mentioned above, or in terms of a Truth Table. A function may be expressed through several algebraic expressions, on account of them being logically equivalent, but there is only one unique truth table for every function.
A Boolean function can be transformed from an algebraic expression into a circuit diagram composed of logic gates connected in a particular structure. Circuit diagram for

Canonical and Standard Forms –
Any binary variable can take one of two forms, or . A boolean function can be expressed in terms of binary variables. If all the binary variables are combined together using the AND operation, then there are a total of combinations since each variable can take two forms.
Each of the combinations is called a minterm or standard product. A minterm is represented by where is the decimal equivalent of the binary number the minterm is designated.
Important Note – In a minterm, the binary variable is un-primed if the variable is 1 and it is primed if the variable is 0 i.e. if the minterm is then that means and .
For example, for a boolean function in two variables the minterms are –

In a similar way, if the variables are combined together with OR operation, then the term obtained is called a maxterm or standard sum. A maxterm is represented by where is the decimal equivalent of the binary number the maxterm is designated.

Important Note – In a maxterm, the binary variable is un-primed if the variable is 0 and it is primed if the variable is 1 i.e. if the maxterm is then that means and .
For example, for a boolean function in two variables the maxterms are –

Minterms and Maxterms for function in 3 variables –

Relation between Minterms and Maxterms – Each minterm is the complement of it’s corresponding maxterm.
For example, for a boolean function in two variables –

Constructing Boolean Functions – Now that we know what minterms and maxterms are, we can use them to construct boolean expressions.

“A Boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a 1 in the function and then taking the OR of all those terms.”

For example, consider two functions and with the following truth tables –

The function is 1 for the following combinations of – 001,100,111
The corresponding minterms are- , , .
Therefore the algebraic expression for is-

Similarly, the algebraic expression for is-

If we use De Morgans Law on and all 1’s become 0 and all 0’s become 1. Therefore we get-

On using De Morgans Law again-

and

We can conclude from the above that boolean functions can be expressed as a sum of minterms or a product of maxterms.

• Example 1 – Express the following boolean expression in SOP and POS forms-
• Solution – The expression can be transformed into SOP form by adding missing variables in each term by multiplying by where is the missing variable.
It follows from the fact that –

On rearranging the minterms in ascending order

If we want the POS form, we can double negate the SOP form as stated above to get-

The SOP and POS forms have a short notation of representation-

Standard Forms –
Canonical forms are basic forms obtained from the truth table of the function. These forms are usually not used to represent the function as they are cumbersome to write and it is preferable to represent the function in the least number of literals possible.
There are two types of standard forms –

1. Sum of Products(SOP)- A boolean expression involving AND terms with one or more literals each, OR’ed together.
2. Product of Sums(POS) A boolean expression involving OR terms with one or more literals each, AND’ed together, e.g.

GATE CS Corner Questions

Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.

Digital Design 5th Edition, by Morris Mano and Michael Ciletti

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## 1.3: Basic Classes of Functions - Mathematics

I'm putting this on the web because some students might find it interesting. It could easily be mentioned in many undergraduate math courses, though it doesn't seem to appear in most textbooks used for those courses. None of this material was discovered by me. -- ES

You should know that the solution of ax 2 +bx+c=0 is

There is an analogous formula for polynomials of degree three: The solution of ax 3 +bx 2 +cx+d=0 is

(A formula like this was first published by Cardano in 1545.) Or, more briefly,

where p = -b/(3a), q = p 3 + (bc-3ad)/(6a 2 ), r = c/(3a)

But I do not recommend that you memorize these formulas.

Aside from the fact that it's too complicated, there are other reasons why we don't teach this formula to calculus students. One reason is that we're trying to avoid teaching them about complex numbers. Complex numbers (i.e., treating points on the plane as numbers) are a more advanced topic, best left for a more advanced course. But then the only numbers we're allowed to use in calculus are real numbers (i.e., the points on the line). That imposes some restrictions on us --- for instance, we can't take the square root of a negative number. Now, Cardan's formula has the drawback that it may bring such square roots into play in intermediate steps of computation, even when those numbers do not appear in the problem or its answer.

For instance, consider the cubic equation x 3 -15x-4=0. (This example was mentioned by Bombelli in his book in 1572.) That problem has real coefficients, and it has three real roots for its answers. (Hint: One of the roots is a small positive integer now can you find all three roots?) But if we apply Cardano's formula to this example, we use a=1, b=0, c=-15, d=-4, and we find that we need to take the square root of -109 in the resulting computation. Ultimately, the square roots of negative numbers would cancel out later in the computation, but that computation can't be understood by a calculus student without additional discussion of complex numbers.

There is also an analogous formula for polynomials of degree 4, but it's much worse to write down I won't even try here.

## 1.3: Basic Classes of Functions - Mathematics

Finding Domain and Range

· Find the domain of a square root function.

· Find the domain and range of a function from the algebraic form.

Functions are a correspondence between two sets, called the domain and the range. When defining a function, you usually state what kind of numbers the domain (x) and range (f(x)) values can be. But even if you say they are real numbers, that doesn’t mean that all real numbers can be used for x. It also doesn’t mean that all real numbers can be function values, f(x). There may be restrictions on the domain and range. The restrictions partly depend on the type of function.

In this topic, all functions will be restricted to real number values. That is, only real numbers can be used in the domain, and only real numbers can be in the range.

Restricting the domain

There are two main reasons why domains are restricted.

· You can’t take the square (or other even) root of a negative number, as the result will not be a real number.

In what kind of functions would these two issues occur?

Division by 0 could happen whenever the function has a variable in the denominator of a rational expression. That is, it’s something to look for in rational functions. Look at these examples, and note that “division by 0” doesn’t necessarily mean that x is 0!

If x = 0, you would be dividing by 0, so x ≠ 0.

If x = 3, you would be dividing by 0, so x ≠ 3.

Although you can simplify this function to

f (x) = 2, when x = 1 the original function would include division by 0. So x ≠ 1.

Both x = 1 and x = −1 would make the denominator 0. Again, this function can be simplified to , but when x = 1 or x = −1 the original function would include division by 0, so x ≠ 1 and x ≠ −1.

This is an example with no domain restrictions, even though there is a variable in the denominator. Since x 2 ≥ 0, x 2 + 1 can never be 0. The least it can be is 1, so there is no danger of division by 0.

Square roots of negative numbers could happen whenever the function has a variable under a radical with an even root. Look at these examples, and note that “square root of a negative variable” doesn’t necessarily mean that the value under the radical sign is negative! For example, if x = −4, then −x = −(−4) = 4, a positive number.

Restrictions to the Domain

If x < 0, you would be taking the square root of a negative number, so x ≥ 0.

If x < −10, you would be taking the square root of a negative number, so x ≥ −10.

When is -x negative? Only when x is positive. (For example, if x = − 3, then − x = 3. If x = 1, then − x = − 1.) This means x ≤ 0.

x 2 – 1 must be positive, x 2 – 1 > 0.

So x 2 > 1. This happens only when x is greater than 1 or less than − 1: x ≤ − 1 or x ≥ 1.

There are no domain restrictions, even though there is a variable under the radical. Since

x 2 ≥ 0, x 2 + 10 can never be negative. The least it can be is 10, so there is no danger of taking the square root of a negative number.

Domains can be restricted if:

· the function is a rational function and the denominator is 0 for some value or values of x.

· the function is a radical function with an even index (such as a square root), and the radicand can be negative for some value or values of x.

Remember, here the range is restricted to all real numbers. The range is also determined by the function and the domain. Consider these graphs, and think about what values of y are possible, and what values (if any) are not. In each case, the functions are real-valued—that is, x and f(x) can only be real numbers.

Quadratic function, f(x) = x 2 – 2x – 3

Remember the basic quadratic function: f(x) = x 2 must always be positive, so f(x) ≥ 0 in that case. In general, quadratic functions always have a point with a maximum or greatest value (if it opens down) or a minimum or least value (it if opens up, like the one above). That means the range of a quadratic function will always be restricted to being above the minimum value or below the maximum value. For the function above, the range is f(x) ≥ −4.

Other polynomial functions with even degrees will have similar range restrictions. Polynomial functions with odd degrees, like f(x) = x 3 , will not have restrictions.

Square root functions look like half of a parabola, turned on its side. The fact that the square root portion must always be positive restricts the range of the basic function, , to only positive values. Changes to that function, such as the negative in front of the radical or the subtraction of 2, can change the range. The range of the function above is f(x) ≤ −2.

Rational function, f(x) =

Rational functions may seem tricky. There is nothing in the function that obviously restricts the range. However, rational functions have asymptotes—lines that the graph will get close to, but never cross or even touch. As you can see in the graph above, the domain restriction provides one asymptote, x = 6. The other is the line y = 1, which provides a restriction to the range. In this case, there are no values of x for which f(x) = 1. So, the range for this function is all real numbers except 1.

Determining Domain and Range

Finding domain and range of different functions is often a matter of asking yourself, what values can this function not have?

What are the domain and range of the real-valued function f(x) = x + 3?

This is a linear function. Remember that linear functions are lines that continue forever in each direction.

Any real number can be substituted for x and get a meaningful output. For any real number, you can always find an x value that gives you that number for the output. Unless a linear function is a constant, such as f(x) = 2, there is no restriction on the range.

The domain and range are all real numbers.

What are the domain and range of the real-valued function f(x) = −3x 2 + 6x + 1?

This is a quadratic function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

Because the coefficient of x 2 is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value.

The vertex, or turning point, is at (1, 4). From the graph, you can see that f(x) ≤ 4.

The domain is all real numbers, and the range is all real numbers f(x) such that f(x) ≤ 4.

You can check that the vertex is indeed at (1, 4). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same y value. The vertex must lie on the line of reflection, because it’s the only point that does not have a mirror image!

In the previous example, notice that when x = 2 and when x = 0, the function value is 1. (You can verify this by evaluating f(2) and f(0).) That is, both (2, 1) and (0, 1) are on the graph. The line of reflection here is x = 1, so the vertex must be at the point (1, f(1)). Evaluating f(1) gives f(1) = 4, so the vertex is at (1, 4).

What are the domain and range of the real-valued function ?

This is a radical function. The domain of a radical function is any x value for which the radicand (the value under the radical sign) is not negative. That means x + 5 ≥ 0, so x ≥ −5.

Since the square root must always be positive or 0, . That means .

The domain is all real numbers x where x ≥ −5, and the range is all real numbers f(x) such that f(x) ≥ −2.

What are the domain and range of the real-valued function ?

This is a rational function. The domain of a rational function is restricted where the denominator is 0. In this case, x + 2 is the denominator, and this is 0 only when x = −2.

For the range, create a graph using a graphing utility and look for asymptotes:

One asymptote, a vertical asymptote, is at x =−2, as you should expect from the domain restriction. The other, a horizontal asymptote, appears to be around y = 3. (In fact, it is indeed y = 3.)

The domain is all real numbers except −2, and the range is all real numbers except 3.

You can check the horizontal asymptote, y = 3. Is it possible for to be equal to 3? Write an equation and try to solve it.

Since the attempt to solve ends with a false statement—0 cannot be equal to 6!—the equation has no solution. There is no value of x for which , so this proves that the range is restricted.

Find the domain and range of the real-valued function f(x) = x 2 + 7.

A) The domain is all real numbers and the range is all real numbers f(x) such that

B) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers f(x) such that f(x) ≥ 7.

C) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers.

D) The domain and range are all real numbers.

A) The domain is all real numbers and the range is all real numbers f(x) such that

Correct. Quadratic functions have no domain restrictions. Since x 2 ≥ 0, x 2 + 7 ≥ 7.

B) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers f(x) such that f(x) ≥ 7.

Incorrect. Negative values can be used for x. The correct answer is: The domain is all real numbers and the range is all real numbers f(x) such that f(x) 7.

C) The domain is all real numbers x such that x ≥ 0 and the range is all real numbers.

Incorrect. Negative values can be used for x, but the range is restricted because x 2 ≥ 0. The correct answer is: The domain is all real numbers and the range is all real numbers f(x) such that f(x) 7.

D) The domain and range are all real numbers.

Incorrect. While it’s true that quadratic functions have no domain restrictions, the range is restricted because x 2 ≥ 0. The correct answer is: The domain is all real numbers and the range is all real numbers f(x) such that f(x) 7.

Although a function may be given as “real valued,” it may be that the function has restrictions to its domain and range. There may be some real numbers that can’t be part of the domain or part of the range. This is particularly true with rational and radical functions, which can have restrictions to domain, range, or both. Other functions, such as quadratic functions and polynomial functions of even degree, also can have restrictions to their range.

## Shifting and scaling a function

We have a function and with some simple operations we can shift the function along the x- and y-axes and also scale it (shrink and enlarge it).

Let us use a polynomial of second degree for this example: .

The graph of this function looks like this:

We can add some constants to the functions that will allow us to shift and scale it: . In our case we start by letting and be 1, and and be 0, which gives us the same .

## 1.3: Basic Classes of Functions - Mathematics

· Understand unit circle, reference angle, terminal side, standard position.

· Find the exact trigonometric function values for angles that measure 30°, 45°, and 60° using the unit circle.

· Find the exact trigonometric function values of any angle whose reference angle measures 30°, 45°, or 60°.

· Determine the quadrants where sine, cosine, and tangent are positive and negative.

Mathematicians create definitions because they have a use in solving certain kinds of problems. For example, the six trigonometric functions were originally defined in terms of right triangles because that was useful in solving real-world problems that involved right triangles, such as finding angles of elevation. The domain, or set of input values, of these functions is the set of angles between 0° and 90°. You will now learn new definitions for these functions in which the domain is the set of all angles. The new functions will have the same values as the original functions when the input is an acute angle. In a right triangle you can only have acute angles, but you will see the definition extended to include other angles.

One use for these new functions is that they can be used to find unknown side lengths and angle measures in any kind of triangle. These new functions can be used in many situations that have nothing to do with triangles at all.

Before looking at the new definitions, you need to become familiar with the standard way that mathematicians draw and label angles.

From geometry, you know that an angle is formed by two rays. The rays meet at a point called a vertex.

In trigonometry, angles are placed on coordinate axes. The vertex is always placed at the origin and one ray is always placed on the positive x-axis. This ray is called the initial side of the angle. The other ray is called the terminal side of the angle. This positioning of an angle is called standard position. The Greek letter theta ( ) is often used to represent an angle measure. Two angles in standard position are shown below.

When an angle is drawn in standard position, it has a direction. Notice that there are little curved arrows in the above drawing. The one on the left goes counterclockwise and is defined to be a positive angle. The one on the right goes clockwise and is defined to be a negative angle. If you used a protractor to measure the angles, you would get 50° in both cases. We refer to the first one as a 50° angle, and we refer to the second one as a angle.

Why would you even have negative angles? As with all definitions, it is a matter of convenience. A spaceship in a circular orbit around Earth’s equator could be traveling in either of two directions. So you could say that it traveled through a angle to indicate that it went in the opposite direction of a spaceship that went through a 50° angle. Why is counterclockwise positive? This is just a convention—something that mathematicians have agreed on—because one way has to be positive and the other way negative.

To see how positive angles result from counterclockwise rotation and negative angles result from clockwise rotation, try the interactive exercise below. Either enter an angle measure in the box labeled “Angle” and hit enter or use the slider to move the terminal side of angle θ through the quadrants.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Draw a 160° angle in standard position.

The angle is positive, so you start at the x-axis and go 160° counterclockwise.

Draw a angle in standard position.

The angle is negative, so you start at the x-axis and go 200° clockwise. Remember that 180° is a straight line. That will bring you to the negative x-axis, and then you have to go 20° farther.

Notice that the terminal sides in the two examples above are the same, but they represent different angles. Such pairs of angles are said to be coterminal angles.

For each angle drawn in standard position, there is a related angle known as a reference angle. This is the angle formed by the terminal side and the x-axis. The reference angle is always considered to be positive, and has a value anywhere from 0° to 90°. Two angles are shown below in standard position.

You can see that the terminal side of the 135° angle and the x-axis form a 45° angle (this is because the two angles must add up to 180°). This 45° angle, shown in red, is the reference angle for 135°. The terminal side of the 205° angle and the x-axis form a 25° angle. It is 25° because . This 25° angle, shown in red, is the reference angle for 205°.

Here are two more angles in standard position.

The terminal side of the 300° angle and the x-axis form a 60° angle (this is because the two angles must add up to 360°). This 60° angle, shown in red, is the reference angle for 300°. The terminal side of the 90° angle and the x-axis form a 90° angle. The reference angle is the same as the original angle in this case. In fact, any angle from 0° to 90° is the same as its reference angle.

What is the reference angle for 100°?

The terminal side is in Quadrant II. The original angle and the reference angle together form a straight line along the x-axis, so their sum is 180 °.

## 1.3: Basic Classes of Functions - Mathematics

Natasha Glydon

Both doctors and nurses use math every day while providing health care for people around the world. Doctors and nurses use math when they write prescriptions or administer medication. Medical professionals use math when drawing up statistical graphs of epidemics or success rates of treatments. Math applies to x-rays and CAT scans. Numbers provide an abundance of information for medical professionals. It is reassuring for the general public to know that our doctors and nurses have been properly trained by studying mathematics and its uses for medicine.

Prescriptions and Medication

Regularly, doctors write prescriptions to their patients for various ailments. Prescriptions indicate a specific medication and dosage amount. Most medications have guidelines for dosage amounts in milligrams (mg) per kilogram (kg). Doctors need to figure out how many milligrams of medication each patient will need, depending on their weight. If the weight of a patient is only known in pounds, doctors need to convert that measurement to kilograms and then find the amount of milligrams for the prescription. There is a very big difference between mg/kg and mg/lbs, so it is imperative that doctors understand how to accurately convert weight measurements. Doctors must also determine how long a prescription will last. For example, if a patient needs to take their medication, say one pill, three times a day. Then one month of pills is approximately 90 pills. However, most patients prefer two or three month prescriptions for convenience and insurance purposes. Doctors must be able to do these calculations mentally with speed and accuracy.

Doctors must also consider how long the medicine will stay in the patient&rsquos body. This will determine how often the patient needs to take their medication in order to keep a sufficient amount of the medicine in the body. For example, a patient takes a pill in the morning that has 50mg of a particular medicine. When the patient wakes up the next day, their body has washed out 40% of the medication. This means that 20mg have been washed out and only 30mg remain in the body. The patient continues to take their 50mg pill each morning. This means that on the morning of day two, the patient has the 30mg left over from day one, as well as another 50mg from the morning of day two, which is a total of 80mg. As this continues, doctors must determine how often a patient needs to take their medication, and for how long, in order to keep enough medicine in the patient&rsquos body to work effectively, but without overdosing.

The amount of medicine in the body after taking a medication decreases by a certain percentage in a certain time (perhaps 10% each hour, for example). This percentage decrease can be expressed as a rational number, 1/10. Hence in each hour, if the amount at the end of the hour decreases by 1/10 then the amount remaining is 9/10 of the amount at the beginning of the hour. This constant rational decrease creates a geometric sequence. So, if a patient takes a pill that has 200mg of a certain drug, the decrease of medication in their body each hour can be seen in the folowing table. The Start column contains the number of mg of the drug remaining in the system at the start of the hour and the End column contains the number of mg of the drug remaining in the system at the end of the hour.

Hour Start End
1 200 9/10 x 200 = 180
2 180 9/10 x 180 = 162
3 162 9/10 x 162 = 145.8
. . .

The sequence of numbers shown above is geometric because there is a common ratio between terms, in this case 9/10. Doctors can use this idea to quickly decide how often a patient needs to take their prescribed medication.

Nurses also use ratios and proportions when administering medication. Nurses need to know how much medicine a patient needs depending on their weight. Nurses need to be able to understand the doctor&rsquos orders. Such an order may be given as: 25 mcg/kg/min. If the patient weighs 52kg, how many milligrams should the patient receive in one hour? In order to do this, nurses must convert micrograms (mcg) to milligrams (mg). If 1mcg = 0.001mg, we can find the amount (in mg) of 25mcg by setting up a proportion.

By cross-multiplying and dividing, we see that 25mcg = 0.025mg. If the patient weighs 52kg, then the patient receives 0.025(52) = 1.3mg per minute. There are 60 minutes in an hour, so in one hour the patient should receive 1.3(60) = 78mg. Nurses use ratios and proportions daily, as well as converting important units. They have special &ldquoshortcuts&rdquo they use to do this math accurately and efficiently in a short amount of time.

Numbers give doctors much information about a patient&rsquos condition. White blood cell counts are generally given as a numerical value between 4 and 10. However, a count of 7.2 actually means that there are 7200 white blood cells in each drop of blood (about a microlitre). In much the same way, the measure of creatinine (a measure of kidney function) in a blood sample is given as X mg per deciliter of blood. Doctors need to know that a measure of 1.3 could mean some extent of kidney failure. Numbers help doctors understand a patient&rsquos condition. They provide measurements of health, which can be warning signs of infection, illness, or disease.

In terms of medicine and health, a person&rsquos Body Mass Index (BMI) is a useful measure. Your BMI is equal to your weight in pounds, times 704.7, divided by the square of your height in inches. This method is not always accurate for people with very high muscle mass because the weight of muscle is greater than the weight of fat. In this case, the calculated BMI measurement may be misleading. There are special machines that find a person&rsquos BMI. We can find the BMI of a 145-pound woman who is 5&rsquo6&rdquo tall as follows.

First, we need to convert the height measurement of 5&rsquo6&rdquo into inches, which is 66&rdquo. Then, the woman&rsquos BMI would be:

This is a normal Body Mass Index. A normal BMI is less than 25. A BMI between 25 and 29.9 is considered to be overweight and a BMI greater than 30 is considered to be obese. BMI measurements give doctors information about a patient&rsquos health. Doctor&rsquos can use this information to suggest health advice for patients. The image below is a BMI table that gives an approximation of health and unhealthy body mass indexes.

One of the more advanced ways that medical professionals use mathematics is in the use of CAT scans. A CAT scan is a special type of x-ray called a Computerized Axial Tomography Scan. A regular x-ray can only provide a two-dimensional view of a particular part of the body. Then, if a smaller bone is hidden between the x-ray machine and a larger bone, the smaller bone cannot be seen. It is like a shadow.

It is much more beneficial to see a three dimensional representation of the body&rsquos organs, particularly the brain. CAT scans allow doctors to see inside the brain, or another body organ, with a three dimensional image. In a CAT scan, the x-ray machine moves around the body scanning the brain (or whichever body part is being scanned) from hundreds of different angles. Then, a computer takes all the scans together and creates a three dimensional image. Each time the x-ray machine makes a full revolution around the brain, the machine is producing an image of a thin slice of the brain, starting at the top of the head and moving down toward the neck. The three-dimensional view created by the CAT scan provides much more information to doctors that a simple two-dimensional x-ray.

Mathematics plays a crucial role in medicine and because people&rsquos lives are involved, it is very important for nurses and doctors to be very accurate in their mathematical calculations. Numbers provide information for doctors, nurses, and even patients. Numbers are a way of communicating information, which is very important in the medical field.

Another application of mathematics to medicine involves a lithotripter. This is a medical device that uses a property of an ellipse to treat gallstones and kidney stones. To learn more, visit the Lithotripsy page.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.

## 1.3: Basic Classes of Functions - Mathematics

A power function is a function of the form,

where a &ne 0 is a constant and p is a real number. Some examples of power functions include:

Root functions, such as are examples of power functions. Graphically, power functions can resemble exponential or logarithmic functions for some values of x. However, as x gets very large, power functions and exponential or logarithmic functions begin to diverge from one another. An exponentially growing function will overtake a growing power function for large values of x. On the other hand, growing power functions will overtake logarithmic functions for large values of x.

Domain and Range

The domain of a power function depends on the value of the power p. We will look at each case separately.

1. p is a non-negative integer

The domain is all real numbers (i.e. (&minus &infin,&infin)).

2. p is a negative integer

The domain is all real numbers not including zero (i.e. (&minus&infin, 0) &cup (0,&infin) or <x|x &ne 0>). We will revisit this case when we study rational functions.

3. p is a rational number expressed in lowest terms as r / s and s is even

A. p > 0

The domain is non-negative real numbers (i.e. [0,&infin) or <x|x &ge 0>).

B. p < 0

The domain is positive real numbers (i.e. (0,&infin) or <x|x > 0>).

4. p is a rational number expressed in lowest terms as r / s and s is odd

A. p > 0

The domain is all real numbers.

B. p < 0

The domain is all real numbers not including zero.

5. p is an irrational number

A. p > 0

The domain is all non-negative real numbers.

B. p < 0

The domain is all positive real numbers.

In the next section we will study the graphs of power functions.