2.1: Introduction to Functions

Our development of the function concept is a modern one, but quite quick, particularly in light of the fact that today’s definition took over 300 years to reach its present state. We begin with the definition of a relation.

Relations

We use the notation (2, 4) to denote what is called an ordered pair. If you think of the positions taken by the ordered pairs (4, 2) and (2, 4) in the coordinate plane (see Figure (PageIndex{1})), then it is immediately apparent why order is important. The ordered pair (4, 2) is simply not the same as the ordered pair (2, 4).

The first element of an ordered pair is called its abscissa. The second element of an ordered pair is called its ordinate. Thus, for example, the abscissa of (4, 2) is 4, while the ordinate of (4, 2) is 2.

Definition: Relation

A collection of ordered pairs is called a relation.

For example, the collection of ordered pairs [R={(0,1),(0,2),(3,4)}] is a relation.

Definition: Domain

The domain of a relation is the collection of all abscissas of each ordered pair.

Thus, the domain of the relation R in (2) is [ ext { Domain }={0,3}]

Note that we list each abscissa only once.

Definition: Range

The range of a relation is the collection of all ordinates of each ordered pair.

Thus, the range of the relation R in (2) is [ ext { Range }={1,2,4}]

Let’s look at an example.

Example (PageIndex{1})

Consider the relation T defined by [T={(1,2),(3,2),(4,5)}]

What are the domain and range of this relation?

Solution

The domain is the collection of abscissas of each ordered pair. Hence, the domain of (T) is

[ ext { Domain }={1,3,4}]

The range is the collection of ordinates of each ordered pair. Hence, the range of (T) is

[ ext { Range }={2,5}]

Note that we list each ordinate in the range only once.

In Example (PageIndex{1}), the relation is described by listing the ordered pairs. This is not the only way that one can describe a relation. For example, a graph certainly represents a collection of ordered pairs.

Example (PageIndex{2})

Consider the graph of the relation (S) shown in Figure (PageIndex{2}).

What are the domain and range of the relation (S)?

Solution

There are five ordered pairs (points) plotted in Figure (PageIndex{2}). They are

[S={(1,2),(2,1),(2,4),(3,3),(4,4)} onumber]

Therefore, the relation S has Domain = {1, 2, 3, 4} and Range = {1, 2, 3, 4}. In the case of the range, note how we’ve sorted the ordinates of each ordered pair in ascending order, taking care not to list any ordinate more than once.

Functions

A function is a very special type of relation. We begin with a formal definition.

Definition: Function

A relation is a function if and only if each object in its domain is paired with one and only one object in its range.

This is not an easy definition, so let’s take our time and consider a few examples. Consider, if you will, the relation R in (2), repeated here again for convenience.

[R={(0,1),(0,2),(3,4)}]

The domain is {0, 3} and the range is {1, 2, 4}. Note that the number 0 in the domain of R is paired with two numbers from the range, namely, 1 and 2. Therefore, R is not a function.

There is a construct, called a mapping diagram, which can be helpful in determining whether a relation is a function. To craft a mapping diagram, first list the domain on the left, then the range on the right, then use arrows to indicate the ordered pairs in your relation, as shown in Figure (PageIndex{3}).

It’s clear from the mapping diagram in Figure (PageIndex{3}) that the number 0 in the domain is being paired (mapped) with two different range objects, namely, 1 and 2. Thus, R is not a function.

Let’s look at another example.

Example (PageIndex{3})

Is the relation described in Example (PageIndex{1}) a function?

Solution

First, let’s repeat the listing of the relation T here for convenience.

[T={(1,2),(3,2),(4,5)} onumber]

Next, construct a mapping diagram for the relation T. List the domain on the left, the range on the right, then use arrows to indicate the pairings, as shown in Figure (PageIndex{4}).

From the mapping diagram in Figure (PageIndex{4}), we can see that each domain object on the left is paired (mapped) with exactly one range object on the right. Hence, the relation T is a function.

Let’s look at another example.

Example (PageIndex{4})

Is the relation of Example (PageIndex{2}), pictured in Figure (PageIndex{2}), a function?

Solution

First, we repeat the graph of the relation from Example (PageIndex{2}) here for convenience. This is shown in Figure (PageIndex{5})(a). Next, we list the ordered pairs of the relation S.

[S={(1,2),(2,1),(2,4),(3,3),(4,4)} onumber]

Then we create a mapping diagram by first listing the domain on the left, the range on the right, then using arrows to indicate the pairings, as shown in Figure (PageIndex{5})(b).

Each object in the domain of S gets mapped to exactly one range object with one exception. The domain object 2 is paired with two range objects, namely, 1 and 4. Consequently, S is not a function.

This is a good point to summarize what we’ve learned about functions thus far.

Summary

A function consists of three parts:

1. a set of objects which mathematicians call the domain,
2. a second set of objects which mathematicians call the range,
3. and a rule that describes how to assign a unique range object to each object in the domain.

The rule can take many forms. For example, we can use sets of ordered pairs, graphs, and mapping diagrams to describe the function. In the sections that follow, we will explore other ways of describing a function, for example, through the use of equations and simple word descriptions.

Function Notation

We’ve used the word “mapping” several times in the previous examples. This is not a word to be taken lightly; it is an important concept. In the case of the mapping diagram in Figure (PageIndex{5})(b), we would say that the number 1 in the domain of S is “mapped” (or “sent”) to the number 2 in the range of S.

There are a number of different notations we could use to indicate that the number 1 in the domain is “mapped” or “sent” to the number 2 in the range. One possible notation is

[S : 1 longrightarrow 2]

which we would read as follows: “The relation S maps (sends) 1 to 2.” In a similar vein, we see in Figure (PageIndex{5})(b) that the domain objects 3 and 4 are mapped (sent) to the range objects 3 and 4, respectively. In symbols, we would write

[egin{array}{l}{S : 3 longrightarrow 3, ext { and }} {S : 4 longrightarrow 4}end{array}]

A difficulty arises when we examine what happens to the domain object 2. There are two possibilities, either

[S : 2 longrightarrow 1] or [S : 2 longrightarrow 4]

Which should we choose? The 1? Or the 4? Thus, S is not well-defined and is not a function, since we don’t know which range object to pair with the domain object 1.

The idea of mapping gives us an alternative way to describe a function. We could say that a function is a rule that assigns a unique object in its range to each object in its domain. Take for example, the function that maps each real number to its square. If we name the function f, then f maps 5 to 25, 6 to 36, −7 to 49, and so on. In symbols, we would write

[f : 5 longrightarrow 25, quad f : 6 longrightarrow 36, quad ext { and } quad f :-7 longrightarrow 49]

In general, we could write

[f : x longrightarrow x^{2}]

Note that each real number x gets mapped to a unique number in the range of f, namely, (x^{2}). Consequently, the function f is well defined. We’ve succeeded in writing a rule that completely defines the function f.

As another example, let’s define a function that takes a real number, doubles it, then adds 3. If we name the function g, then g would take the number 7, double it, then add 3. That is,

[g : 7 longrightarrow 2(7)+3]

Simplifying, (g : 7 longrightarrow 17). Similarly, g would take the number 9, double it, then add 3. That is,

[g : 9 longrightarrow 2(9)+3]

Simplifying, (g : 9 longrightarrow 21). In general, g takes a real number x, doubles it, then adds three. In symbols, we would write

[g : x longrightarrow 2 x+3]

Notice that each real number x is mapped by g to a unique number in its range. Therefore, we’ve again defined a rule that completely defines the function g.

It is helpful to think of a function as a machine. The machine receives input, processes it according to some rule, then outputs a result. Something goes in (input), then something comes out (output). In the case of the function described by the rule (f : x longrightarrow x^{2}), the “f-machine” receives input x, then applies its “square rule” to the input and outputs (x^{2}), as shown in Figure (PageIndex{6})(a). In the case of the function described by the rule (g : x longrightarrow 2 x+3), the “g-machine” receives input x, then applies the rules “double,” then “add 3,” in that order, then outputs (2x + 3), as shown in Figure (PageIndex{6})(b).

Let’s look at another example.

Example (PageIndex{5})

Suppose that f is defined by the following rule. For each real number x,

[f : x longrightarrow x^{2}-2 x-3]

Where does f map the number −3? Is f a function?

Solution

We substitute −3 for x in the rule for f and obtain

[f :-3 longrightarrow(-3)^{2}-2(-3)-3]

Simplifying,

[f :-3 longrightarrow 9+6-3]

or,

[f :-3 longrightarrow 12]

Thus, f maps (sends) the number −3 to the number 12. It should be clear that each real number x will be mapped (sent) to a unique real number, as defined by the rule (f : x longrightarrow x^{2}-2 x-3). Therefore, f is a function.

Let’s look at another example.

Example (PageIndex{6})

Suppose that g is defined by the following rule. For each real number x that is greater than or equal to zero,

[g : x longrightarrow pm sqrt{x}]

Where does g map the number 4? Is g a function?

Solution

Again, we substitute 4 for x in the rule for g and obtain

[g : 4 longrightarrow pm sqrt{4}]

Simplifying,

[g : 4 longrightarrow pm 2]

Thus, g maps (sends) the number 4 to two different objects in its range, namely, 2 and −2. Consequently, g is not well-defined and is not a function.

Let’s look at another example

Example (PageIndex{7})

Suppose that we have functions f and g, defined by

[f : x longrightarrow x^{4}+11 quad ext { and } quad g : x longrightarrow(x+2)^{2}]

Where does g send 5?

Solution

In this example, we see a clear advantage of function notation. Because our functions have distinct names, we can simply reference the name of the function we want our readers to use. In this case, we are asked where the function g sends the number 5, so we substitute 5 for x in

[g : x longrightarrow(x+2)^{2}]

That is,

[g : 5 longrightarrow(5+2)^{2}]

Simplifying, (g : 5 longrightarrow 49).

Modern Notation

Function notation is relatively new, with some of the earliest symbolism first occurring in the 17th century. In a letter to Leibniz (1698), Johann Bernoulli wrote “For denoting any function of a variable quantity x, I rather prefer to use the capital letter having the same name X or the Greek (xi), for it appears at once of what variable it is a function; this relieves the memory.”

Mathematicians are fond of the notation [f : x longrightarrow x^{2}-2 x]

because it conveys a sense of what a function does; namely, it “maps” or “sends” the number x to the number (x^{2}-2 x). This is what functions do, they pair each object in their domain with a unique object in their range. Equivalently, functions “send” each object in their domain to a unique object in their range.

However, in common computational situations, the “arrow” notation can be a bit clumsy, so mathematicians tend to favor a slightly different notation. Instead of writing

[f : x longrightarrow x^{2}-2 x]

mathematicians tend to favor the notation

[f(x)=x^{2}-2 x]

It is important to understand from the outset that these two different notations are equivalent; they represent the same function f, one that pairs each real number x in its domain with the real number (x^{2}-2 x) in its range.

The first notation, (f : x longrightarrow x^{2}-2 x), conveys the sense that the function f is a mapping. If we read this notation aloud, we should pronounce it as “f sends (or maps) x to (x^{2}-2 x).” The second notation, (f(x) = x^{2}-2 x), is pronounced “f of x equals (x^{2}-2 x).”

Note

The phrase “f of x” is unfortunate, as our readers might recall being trained from a very early age to pair the word “of” with the operation of multiplication. For example, 1/2 of 12 is 6, as in (1 / 2 imes 12=6). However, in the context of function notation, even though f(x) is read aloud as “f of x,” it does not mean “f times x.” Indeed, if we remind ourselves that the notation (f(x)=x^{2}-2 x) is equivalent to the notation (f : x longrightarrow x^{2}-2 x), then even though we might say “f of x,” we should be thinking “f sends x” or “f maps x.” We should not be thinking “f times x.”

Now, let’s see how each of these notations operates on the number 5. In the first case, using the “arrow” notation,

[f : x longrightarrow x^{2}-2 x]

To find where f sends 5, we substitute 5 for x as follows.

[f : 5 longrightarrow(5)^{2}-2(5)]

Simplifying,(f : 5 longrightarrow 15). Now, because both notations are equivalent, to compute f(5), we again substitute 5 for x in

[f(x)=x^{2}-2 x]

Thus,

[f(5)=(5)^{2}-2(5)]

Simplifying, (f(5)=15). This result is read aloud as “f of 5 equals 15,” but we want to be thinking “f sends 5 to 15.”

Let’s look at examples that use this modern notation.

Example (PageIndex{8})

Given (f(x)=x^{3}+3 x^{2}-5,) determine (f(-2))

Solution

Simply substitute −2 for x. That is,

[egin{aligned} f(-2) &=(-2)^{3}+3(-2)^{2}-5 &=-8+3(4)-5 &=-8+12-5 &=-1 end{aligned}]

Thus, (f(−2) = −1). Again, even though this is pronounced “f of −2 equals −1,” we still should be thinking “f sends −2 to −1.”

Example (PageIndex{9})

Given [f(x)=frac{x+3}{2 x-5}] determine f(6).

Solution

Simply substitute 6 for x. That is, [egin{aligned} f(6) &=frac{6+3}{2(6)-5} &=frac{9}{12-5} &=frac{9}{7} end{aligned}]

Thus, (f(6) = 9/7). Again, even though this is pronounced “f of 6 equals 9/7,” we should still be thinking “f sends 6 to 9/7.”

Example (PageIndex{10})

Given (f(x)=5 x-3,) determine (f(a+2)).

Solution

If we’re thinking in terms of mapping notation, then [f : x longrightarrow 5 x-3]

Think of this mapping as a “machine.” Whatever we put into the machine, it first is multiplied by 5, then 3 is subtracted from the result, as shown in Figure (PageIndex{7}). For example, if we put a 4 into the machine, then the function rule requires that we multiply input 4 by 5, then subtract 3 from the result. That is,

[f : 4 longrightarrow 5(4)-3]

Simplifying, (f : 4 longrightarrow 17)

Similarly, if we put an a + 2 into the machine, then the function rule requires that we multiply the input a + 2 by 5, then subtract 3 from the result. That is,

[f : a+2 longrightarrow 5(a+2)-3]

Using modern function notation, we would write

[f(a+2)=5(a+2)-3]

Note that this is again a simple substitution, where we replace each occurrence of x in the formula (f(x) = 5x − 3) with the expression a + 2. Finally, use the distributive property to first multiply by 5, then subtract 3.

[egin{aligned} f(a+2) &=5 a+10-3 &=5 a+7 end{aligned}]

We will often need to substitute the result of one function evaluation into a second function for evaluation. Let’s look at an example.

Example (PageIndex{11})

Given two functions defined by (f(x) = 3x + 2) and (g(x) = 5 − 4x), find f(g(2)).

Solution

The nested parentheses in the expression f(g(2)) work in the same manner that they do with nested expressions. The rule is to work the innermost grouping symbols first, proceeding outward as you work. We’ll first evaluate g(2), then evaluate f at the result.

We begin. First, evaluate g(2) by substituting 2 for x in the defining equation (g(x) = 5 − 4x). Note that (g(2) = 5 − 4(2)), then simplify.

[f(g(2))=f(5-4(2))=f(5-8)=f(-3)]

To complete the evaluation, we substitute −3 for x in the defining equation (f(x) = 3x + 2), then simplify.

[f(-3)=3(-3)+2=-9+2=-7]

Hence, (f(g(2))=-7).

It is conventional to arrange the work in one contiguous block, as follows.

[egin{aligned} f(g(2)) &=f(5-4(2)) &=f(-3) &=3(-3)+2 &=-7 end{aligned}]

You can shorten the task even further if you are willing to do the function substitution and simplification in your head. First, evaluate g at 2, then f at the result.

[f(g(2))=f(-3)=-7]

Let’s look at another example of this unique way of combining functions.

Example (PageIndex{12})

Given (f(x) = 5x + 2) and (g(x) = 3 − 2x), evaluate (g(f(a))) and simplify the result.

Solution

We work the inner function evaluation in the expression (g(f(a))) first. Thus, to evaluate f(a), we substitute a for x in the definition (f(x) = 5x + 2) to get

[g(f(a))=g(5 a+2)]

Now we need to evaluate (g(5a + 2)). To do this, we substitute (5a + 2) for x in the definition (g(x) = 3 − 2x) to get

[g(5 a+2)=3-2(5 a+2)]

We can expand this last result and simplify. Thus,

[g(f(a))=3-10 a-4=-10 a-1]

Again, it is conventional to arrange the work in one continuous block, as follows.

[egin{aligned} g(f(a)) &=g(5 a+2) &=3-2(5 a+2) &=3-10 a-4 &=-10 a-1 end{aligned}]

Hence, (g(f(a))=-10 a-1).

Extracting the Domain of a Function

We’ve seen that the domain of a relation or function is the set of all the first coordinates of its ordered pairs. However, if a functional relationship is defined by an equation such as (f(x) = 3x − 4), then it is not practical to list all ordered pairs defined by this relationship. For any real x-value, you get an ordered pair. For example, if x = 5, then (f(5) = 3(5) − 4 = 11), leading to the ordered pair (5, f(5)) or (5, 11). As you can see, the number of such ordered pairs is infinite. For each new x-value, we get another function value and another ordered pair.

Therefore, it is easier to turn our attention to the values of x that yield real number responses in the equation (f(x) = 3x − 4). This leads to the following key idea.

Definition

If a function is defined by an equation, then the domain of the function is the set of “permissible x-values,” the values that produce a real number response defined by the equation.

We sometimes like to say that the domain of a function is the set of “OK x-values to use in the equation.” For example, if we define a function with the rule (f(x) = 3x − 4), it is immediately apparent that we can use any value we want for x in the rule (f(x) = 3x − 4). Thus, the domain of f is all real numbers. We can write that the domain (D=mathbb{R}), or we can use interval notation and write that the domain (D=(-infty, infty)).

It is not the case that x can be any real number in the function defined by the rule (f(x)=sqrt{x}). It is not possible to take the square root of a negative number.2 Therefore, x must either be zero or a positive real number. In set-builder notation, we can describe the domain with (D={x : x geq 0}). In interval notation, we write (D=[0, infty)).

We must also be aware of the fact that we cannot divide by zero. If we define a function with the rule (f(x)=x /(x-3)), we immediately see that x = 3 will put a zero in the denominator. Division by zero is not defined. Therefore, 3 is not in the domain of f. No other x-value will cause a problem. The domain of f is best described with set-builder notation as (D={x : x eq 3}).

Functions Without Formulae

In the previous section, we defined functions by means of a formula, for example, as in

[f(x)=frac{x+3}{2-3 x}]

Euler would be pleased with this definition, for as we have said previously, Euler thought of functions as analytic expressions.

However, it really isn’t necessary to provide an expression or formula to define a function. There are other forms we can use to express a functional relationship: a graph, a table, or even a narrative description. The only thing that is really important is the requirement that the function be well-defined, and by “well-defined,” we mean that each object in the function’s domain is paired with one and only one object in its range.

As an example, let’s look at a special function (pi) on the natural numbers,3 which returns the number of primes less than or equal to a given natural number. For example, the primes less than or equal to the number 23 are 2, 3, 5, 7, 11, 13, 17, 19, and 23, nine numbers in all. Therefore, the number of primes less than or equal to 23 is nine. In symbols, we would write

[pi(23)=9]

Note the absence of a formula in the definition of this function. Indeed, the definition is descriptive in nature, so we might write

[pi(n)= ext { number of primes less than or equal to } n]

The important thing is not how we define this special function π, but the fact that it is well-defined; that is, for each natural number n, there are a fixed number of primes less than or equal to n. Thus, each natural number in the domain of π is paired with one and only one number in its range.

Now, just because our function doesn’t provide an expression for calculating the number of primes less than or equal to a given natural number n, it doesn’t stop mathematicians from seeking such a formula. Euclid of Alexandria (325-265 BC), a Greek mathematician, proved that the number of primes is infinite, but it was the German mathematician and scientist, Johann Carl Friedrich Gauss (1777-1855), who first proposed that the number of primes less than or equal to n can be approximated by the formula

[pi(n) approx frac{n}{ln n}]

where ln n is the “natural logarithm” of n (to be explained in Chapter 9). This approximation gets better and better with larger and larger values of n. The formula was refined by Gauss, who did not provide a proof, and the problem became known as the Prime Number Theorem. It was not until 1896 that Jacques Salomon Hadamard (1865-1963) and Charles Jean Gustave Nicolas Baron de la Vallee Poussin (1866-1962), working independently, provided a proof of the Prime Number Theorem.

2.1: Introduction to Functions

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Arts and culture leaders have a tough but rewarding task: creating and leading sustainable organizations that deliver real social value. There is a lot of competition out there. Being an effective leader means constantly adapting, cleverly using the best tools to reach as many people as possible. This course is designed to help leaders at any level do just that.

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An excellent course with great presentations and very useful and interesting assignments! Thank you!

Definitely an enjoyable course! I have learned a lot and gained confidence for future work activities.

Form Follows Function: Organizational Structures Aligned to Purpose

How do we deliver on value? This week explores form and function of organizations. By the end of the unit, you will clearly understand organization structures – historic and behavioral biases that keep us from changing current structures, frameworks and logic that build better structures. And you will better understand risk - how best to identify and manage it.

This course caters to different user experience levels. Depending on which category you consider yourself to be in, you can expect a different set of course outcomes. Each category contains information that is essential for the next one, so it’s important to do all exercises that are at or below your level of experience.

2.1.2.1. Basic¶

In this category, the course assumes that you have little or no prior experience with theoretical GIS knowledge or the operation of a GIS program.

Limited theoretical background will be provided to explain the purpose of an action you will be performing in the program, but the emphasis is on learning by doing.

When you complete the course, you will have a better concept of the possibilities of GIS, and how to harness their power via QGIS.

2.1.2.2. Intermediate¶

In this category, it is assumed that you have working knowledge and experience of the everyday uses of GIS.

Following the instructions for the beginner level will provide you with familiar ground, as well as to make you aware of the cases where QGIS does things slightly differently from other software you may be used to. You will also learn how to use analysis functions in QGIS.

When you complete the course, you should be comfortable with using QGIS for all of the functions you usually need from a GIS for everyday use.

In this category, the assumption is that you are experienced with GIS, have knowledge of and experience with spatial databases, using data on a remote server, perhaps writing scripts for analysis purposes, etc.

Following the instructions for the other two levels will familiarize you with the approach that the QGIS interface follows, and will ensure that you know how to access the basic functions that you need. You will also be shown how to make use of QGIS’ plugin system, database access system, and so on.

When you complete the course, you should be well-acquainted with the everyday operation of QGIS, as well as its more advanced functions.

2.3 Function calls

Let’s take a closer look at the syntax of R function calls.

The function shown below consists of a name function_name and two arguments, arg1 and arg2 . The arguments may have default values. In this example, arg1 doesn’t have a default value, but arg2 has the default value val2 . Arguments with no default value are required, whereas arguments with a default value are not. These simply take their default value if the a value is not explictly provided.

A function may have many arguments.

= vs. <- for function arguments.

The == operator is used in order to test for equivalance.

Using tab completion for function arguments:

At the R prompt, enter scale( and press TAB .

What are the arguments of the function round() ? Do any have default values?

Look up the rnorm() function in the Help Viewer. What arguments? Any default values?

Do the same for the seq() function.

2.3.1 Nested function calls

Function calls can be nested. This means that the output of one function is passed as input to the next function.

For example: Let’s define a vector, campute its mean and then round to two decimal places:

Function calls are always performed in the same order: from innermost to outermost, e.g. first mean() and then round() .

Lane ORCCA (2020–2021): Open Resources for Community College Algebra

In mathematics, we use functions to model real-life data. In this section, we will learn the definition of a function and related concepts.

Subsection 8.1.1 Introduction to Functions

When working with two variables, we are interested in the relationship between those two variables. For example, consider the two variables of hare population and lynx population in a Canadian forest. If we know the value of one variable, are we able to determine the value of the second variable? If we know that one variable is increasing over time, do we know if the other is increasing or decreasing?

Definition 8.1.2 . Relation.

A is a very general situation between two variables, where having a little bit of information about one variable could tell you something about the other variable. For example, if you know the hare population is high this year, you can say the lynx population is probably increasing. So “hare population” and “lynx population” make a relation. If one of the variables is identified as the “first” variable, the relation's is the set of all values that variable can take. Likewise, the relation's is the set of all values that the second variable can take.

We are not so much concerned with relations in this book. But we are interested in a special type of relation called a function. Informally, a is a device that takes input values for one variable one by one, thinks about them, and gives respective output values one by one for the other variable.

Example 8.1.3 .

Mariana has (5) chickens: Hazel, Yvonne, Georgia, Isabella, and Emma. For the relation “Chicken to Egg Color,” the first variable (the input) is a chicken's name and the second variable (the output) is the color of that chicken's eggs. The relation's domain is the set of all of Mariana's chicken's names, and its range is the set of colors of her chicken's eggs. Figure 8.1.4 shows two inputs and their corresponding outputs.

It would not be convenient to make diagrams like the ones in Figure 8.1.4 for all five chickens. There are too many inputs. Instead, Figure 8.1.5 represents the function graphically in a more concise way. The function's input variable is “chicken name,” and its output variable is “egg color.” Note that we are using the word “variable,” because the chicken names and egg colors vary depending on which individual chicken you choose.

We can also use a set of ordered pairs to represent this function:

where you read the ordered pair left to right, with the first value as an input and the second value as its output.

Definition 8.1.6 . Function.

In mathematics, a function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output.

In Figure 8.1.5, we can see each chicken's name (input) is related to exactly one output, so the relation “Chicken to Egg Color” qualifies as a function. Note that it is irrelevant that multiple inputs might be related to the same output, like in (( ext, ext)) and (( ext, ext) ext<.>) The point is that whichever chicken you are thinking about, you know exactly which color egg it lays.

Subsection 8.1.2 Algebraic Functions and Function Notation

Many functions have specific algebraic formulas to turn an input number into an output number. For example, we know that the equation (y=5x+3) represents (y) as a function of (x ext<,>) because for each (x)-value (input), there is only one (y)-value (output). If we want to determine the value of the output when the input is (2 ext<,>) we'd replace (x) with (2) and find the value of (y ext<:>)

Our end result is that (y=13 ext<.>) Well, (y) is (13 ext<,>) but only in the situation when (x) is (2 ext<.>) In general, for other inputs, (y) is not going to be (13 ext<.>) So the equation (y=13) is lacking in the sense that it is not communicating everything we might want to say. It does not communicate the value of (x) that we used. will allow us to communicate both the input and the output at the same time. It will also allow us to give each function a name, which is helpful when we have multiple functions.

Functions can have names just like variables. The most common function name is (f ext<,>) since “f” stands for “function.” A letter like (f) doesn't stand for a single number though. Instead, it represents an input-output relation like we've been discussing in this section.

We will write equations like (y=f(x) ext<,>) and what we mean is:

the parentheses following the (f) surround the input they do not indicate multiplication

Remark 8.1.7 .

Parentheses have a lot of uses in mathematics. Their use with functions is very specific, and it's important to note that (f) is not being multiplied by anything when we write (f(x) ext<.>) With function notation, the parentheses specifically are just meant to indicate the input by surrounding the input.

Example 8.1.8 .

The expression (f(x)) is read as “(f) of (x ext<,>)” and the expression (f(2)) is read as “(f) of (2 ext<.>)” Be sure to practice saying this correctly while reading.

The expression (f(2)) means that (2) is being treated as an input, and the function (f) is turning it into an output. And then (f(2)) represents that actual output number.

Remark 8.1.9 .

The most common letters used to represent functions are (f,g ext<,>) and (h ext<.>) The most common variables we use are (x ext<,>) (y ext<,>) and (z ext<.>) But we can use any function name and any input and output variable. When dealing with functions in context, it often makes sense to use meaningful function names and variables. For example, if we are modeling temperature of a cup of coffee as a function of time with a function (C ext<,>) we could use (T=C(t) ext<,>) where (T) is the temperature (in degrees Fahrenheit) after (t) minutes.

Subsection 8.1.3 Evaluating Functions

When we determine a function's value for a specific input, this is known as evaluating a function. To do so, we replace the input with the numerical value given and determine the associated output.

When using function notation, instead of writing (5x+3) or (y=5x+3 ext<,>) we often write something like (f(x)=5x+3 ext<.>) We are saying that the rule for function (f) is to use the expression (5x+3 ext<.>) To find (f(2) ext<,>) wherever you see (x) in the formula (f(x)=5x+3 ext<,>) substitute in (2 ext<:>)

Our end result is that (f(2)=13 ext<,>) which tells us that (f) turns (2) into (13 ext<.>) In other words, when the input is (2 ext<,>) the output will be (13 ext<.>)

Let's look at a few more examples.

Example 8.1.10 . Evaluating Functions with Algebraic Formulas.

Find the given function values for a function (f) where (f(x)=2x^2-5x+9 ext<.>)

We find (f(-2)) by replacing all the (x)'s in the formula for (f) with (-2) and then, using the order of operations, simplifying the right side as much as possible.

We find (f(0)) by replacing all the (x)'s in the formula for (f) with (0) and then, using the order of operations, simplifying the right side as much as possible.

We find (f(4)) by replacing all the (x)'s in the formula for (f) with (4) and then, using the order of operations, simplifying the right side as much as possible.

Checkpoint 8.1.11 . Evaluating a Function.

A function may also be described by explicitly listing many inputs and their corresponding outputs in a table.

Example 8.1.12 . Functions given in Table Form.

Temperature readings for Portland, OR, on a given day are recorded in Table 8.1.13. Let (f(x)) be the temperature in degrees Fahrenheit (x) hours after midnight.

Table 8.1.13 . Recorded Temperatures in Portland, OR, on a certain day

 (x ext<,>) hours after midnight (0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (f(x) ext<,>) temperature in °F (45) (44) (42) (42) (43) (44) (45) (48) (49) (50) (53)

What was the temperature at midnight?

Find (f(9) ext<.>) Explain what this function value represents in the context of the problem.

To determine the temperature at midnight, we look in the table where (x=0) and see that the output is (45 ext<.>) Using function notation, we would write:

Thus, at midnight the temperature was 45 °F .

To determine the value of (f(9) ext<,>) we look in the table where (x=9) and read the output:

In context, this means that at 9AM the temperature was 50 °F .

Subsection 8.1.4 Domain and Range

Earlier we defined the domain and range of a relation. We repeat those definitions more formally here, specifically for functions.

Definition 8.1.14 . Domain and Range.

Given a function (f ext<,>) the of (f) is the collection of all valid input values for (f ext<.>) The of (f) is the collection of all possible output values of (f ext<.>)

When working with functions, a common necessary task is to determine the function's domain and range. Also, the ability to identify domain and range is strong evidence that a person really understands the concepts of domain and range.

Example 8.1.15 . Functions Defined by Ordered Pairs.

The function (f) is defined by the ordered pairs

Determine the domain and range of (f ext<.>)

The ordered pairs tell us that (f(1)=2 ext<,>) (f(3)=-2 ext<,>) etc. So the valid input values are (1 ext<,>) (3 ext<,>) (5 ext<,>) (7 ext<,>) and (9 ext<.>) This means the domain is the set (<1,3,5,7,9> ext<.>)

Similarly, the ordered pairs tell us that (2 ext<,>) (-2 ext<,>) (-4 ext<,>) and (6) are possible output values. Notice that the output (2) happened twice, but it only needs to be listed in this collection once. The range of (f) is (<2,-2,-4,6> ext<.>)

Function Notation and Linear Functions

With the definition of a function comes special notation. If we consider each x-value to be the input that produces exactly one output, then we can use the notation

The notation f ( x ) reads “f of x” and should not be confused with multiplication. Most of our study of algebra involves functions, so the notation becomes very useful when performing common tasks. Functions can be named with different letters some common names for functions are g(x), h(x), C(x), and R(x). First, consider nonvertical lines that we know can be expressed using slope-intercept form, y = m x + b . For any real numbers m and b, the equation defines a function, and we can replace y with the new notation f ( x ) as follows:

Therefore, a linear function Any function that can be written in the form f(x) = mx + b. is any function that can be written in the form f ( x ) = m x + b . In particular, we can write the following:

The notation also shows values to evaluate in the equation. If the value for x is given as 8, then we know that we can find the corresponding y-value by substituting 8 in for x and simplifying. Using function notation, this is denoted f ( 8 ) and can be interpreted as follows:

We have f ( 8 ) = 4 . This notation tells us that when x = 8 (the input), the function results in 4 (the output).

Example 4: Given the linear function f ( x ) = − 5 x + 7 , find f ( − 2 ) .

Solution: In this case, f ( − 2 ) indicates that we should evaluate when x = − 2 .

Example 5: Given the linear function f ( x ) = − 5 x + 7 , find x when f ( x ) = 10 .

Solution: In this case, f ( x ) = 10 indicates that the function should be set equal to 10.

Answer: Here x = − 3 5 , and we can write f ( − 3 5 ) = 10 .

Example 6: Given the graph of a linear function g ( x ) ,

a. The notation g ( 2 ) implies that x = 2. Use the graph to determine the corresponding y-value.

b. The notation g ( x ) = 3 implies that the y-value is given as 3. Use the graph to determine the corresponding x-value.

Example 7: Graph the linear function f ( x ) = − 5 3 x + 6 and state the domain and range.

Solution: From the function, we see that b = 6 and thus the y-intercept is (0, 6). Also, we can see that the slope is m = − 5 3 = − 5 3 = r i s e r u n . Starting from the y-intercept, mark a second point down 5 units and right 3 units.

Given any coordinate on the x-axis, we can find a corresponding point on the graph the domain consists of all real numbers. Also, for any coordinate on the y-axis, we can find a point on the graph the range consists of all real numbers.

Answer: Both the domain and range consist of all real numbers R.

Try this! Given the linear function g ( x ) = − x + 5 ,

Key Takeaways

• A relation is any set of ordered pairs. However, in the context of this course, we will be working with sets of ordered pairs (x, y) in the rectangular coordinate system. The set of x-values defines the domain and the set of y-values defines the range.
• Special relations where every x-value (input) corresponds to exactly one y-value (output) are called functions.
• We can easily determine whether an equation represents a function by performing the vertical line test on its graph. If any vertical line intersects the graph more than once, then the graph does not represent a function. In this case, there will be more than one point with the same x-value.

Any nonvertical or nonhorizontal line is a function and can be written using function notation f ( x ) = m x + b . Both the domain and range consist of all real numbers.

• If asked to find f ( a ) , we substitute a in for the variable and then simplify.
• If asked to find x when f ( x ) = a , we set the function equal to a and then solve for x .

Topic Exercises

For each problem below, does the correspondence represent a function?

1. Algebra students to their scores on the first exam.

2. Family members to their ages.

3. Lab computers to their users.

4. Students to the schools they have attended.

5. People to their citizenships.

6. Local businesses to their number of employees.

Determine the domain and range and state whether the relation is a function or not.

2.1: An Overview of Functional Groups

• Contributed by Layne Morsch
• Professor (Chemistry) at University of Illinois Springfield

Functional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity. A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound. Because of their importance in understanding organic chemistry, functional groups have characteristic names that often carry over in the naming of individual compounds incorporating specific groups

As we progress in our study of organic chemistry, it will become extremely important to be able to quickly recognize the most common functional groups, because they are the key structural elements that define how organic molecules react. For now, we will only worry about drawing and recognizing each functional group, as depicted by Lewis and line structures. Much of the remainder of your study of organic chemistry will be taken up with learning about how the different functional groups tend to behave in organic reactions.

Hydrocarbons and halides

We have already seen some examples of very common functional groups: ethene, for example, contains a carbon-carbon double bond. This double bond is referred to, in the functional group terminology, as an alkene.

The carbon-carbon triple bond in ethyne is the simplest example of an alkyne function group.

What about ethane? All we see in this molecule is carbon-hydrogen and carbon-carbon single bonds, so in a sense we can think of ethane as lacking a functional group entirely. However, we do have a general name for this &lsquodefault&rsquo carbon bonding pattern: molecules or parts of molecules containing only carbon-hydrogen and carbon-carbon single bonds are referred to as alkanes.

If the carbon of an alkane is bonded to a halogen, the group is now referred to as a haloalkane (fluoroalkane, chloroalkane, etc.). Chloroform, CHCl3, is an example of a simple haloalkane.

Alcohols and Thiols

We have already seen the simplest possible example of an alcohol functional group in methanol. In the alcohol functional group, a carbon is single-bonded to an OH group (this OH group, by itself, is referred to as a hydroxyl). If the central carbon in an alcohol is bonded to only one other carbon, we call the group a primary alcohol. In secondary alcohols and tertiary alcohols, the central carbon is bonded to two and three carbons, respectively. Methanol, of course, is in class by itself in this respect.

The sulfur analog of an alcohol is called a thiol (the prefix thio, derived from the Greek, refers to sulfur).

In an ether functional group, a central oxygen is bonded to two carbons. Below are the line and Lewis structures of diethyl ether, a common laboratory solvent and also one of the first medical anaesthesia agents.

In sulfides, the oxygen atom of an ether has been replaced by a sulfur atom.

Amines and Phosphates

Ammonia is the simplest example of a functional group called amines. Just as there are primary, secondary, and tertiary alcohols, there are primary, secondary, and tertiary amines.

One of the most important properties of amines is that they are basic, and are readily protonated to form ammonium cations.

Phosphorus is a very important element in biological organic chemistry, and is found as the central atom in the phosphate group. Many biological organic molecules contain phosphate, diphosphate, and triphosphate groups, which are linked to a carbon atom by the phosphate ester functionality.

Because phosphates are so abundant in biological organic chemistry, it is convenient to depict them with the abbreviation 'P'. Notice that this 'P' abbreviation includes the oxygen atoms and negative charges associated with the phosphate groups.

Carbonyl Containing Functional Groups

Aldehydes and Ketones

There are a number of functional groups that contain a carbon-oxygen double bond, which is commonly referred to as a carbonyl. Ketones and aldehydes are two closely related carbonyl-based functional groups that react in very similar ways. In a ketone, the carbon atom of a carbonyl is bonded to two other carbons. In an aldehyde, the carbonyl carbon is bonded on one side to a hydrogen, and on the other side to a carbon. The exception to this definition is formaldehyde, in which the carbonyl carbon has bonds to two hydrogens.

Molecules with carbon-nitrogen double bonds are called imines, or Schiff bases.

Carboxylic acids and acid derivatives

If a carbonyl carbon is bonded on one side to a carbon (or hydrogen) and on the other side to a heteroatom (in organic chemistry, this term generally refers to oxygen, nitrogen, sulfur, or one of the halogens), the functional group is considered to be one of the &lsquocarboxylic acid derivatives&rsquo, a designation that describes a grouping of several functional groups. The eponymous member of this grouping is the carboxylic acid functional group, in which the carbonyl is bonded to a hydroxyl (OH) group.

As the name implies, carboxylic acids are acidic, meaning that they are readily deprotonated to form the conjugate base form, called a carboxylate (much more about carboxylic acids in the acid-base chapter!).

In amides, the carbonyl carbon is bonded to a nitrogen. The nitrogen in an amide can be bonded either to hydrogens, to carbons, or to both. Another way of thinking of an amide is that it is a carbonyl bonded to an amine.

In esters, the carbonyl carbon is bonded to an oxygen which is itself bonded to another carbon. Another way of thinking of an ester is that it is a carbonyl bonded to an alcohol. Thioesters are similar to esters, except a sulfur is in place of the oxygen.

In an acyl phosphate, the carbonyl carbon is bonded to the oxygen of a phosphate, and in an acid chloride, the carbonyl carbon is bonded to a chlorine.

Finally, in a nitrile group, a carbon is triple-bonded to a nitrogen. Nitriles are also often referred to as cyano groups.

A single compound often contains several functional groups. The six-carbon sugar molecules glucose and fructose, for example, contain aldehyde and ketone groups, respectively, and both contain five alcohol groups (a compound with several alcohol groups is often referred to as a &lsquopolyol&rsquo).

Capsaicin, the compound responsible for the heat in hot peppers, contains phenol, ether, amide, and alkene functional groups.

The male sex hormone testosterone contains ketone, alkene, and secondary alcohol groups, while acetylsalicylic acid (aspirin) contains aromatic, carboxylic acid, and ester groups.

While not in any way a complete list, this section has covered most of the important functional groups that we will encounter in biological and laboratory organic chemistry. The table on the inside back cover provides a summary of all of the groups listed in this section, plus a few more that will be introduced later in the text.

Problems

1: Identify the functional groups in the following organic compounds. State whether alcohols and amines are primary, secondary, or tertiary.

2: Draw one example each (there are many possible correct answers) of compounds fitting the descriptions below, using line structures. Be sure to designate the location of all non-zero formal charges. All atoms should have complete octets (phosphorus may exceed the octet rule).

a) a compound with molecular formula C6H11NO that includes alkene, secondary amine, and primary alcohol functional groups

b) an ion with molecular formula C3H5O6P 2- that includes aldehyde, secondary alcohol, and phosphate functional groups.

c) A compound with molecular formula C6H9NO that has an amide functional group, and does not have an alkene group.

2.1: System Poles and Zeros

• Contributed by Kamran Iqbal
• Professor (Systems Engineering) at University of Arkansas at Little Rock

System Poles and Zeros

The transfer function, (G(s)), is a rational function in the Laplace transform variable, (s). It is expressed as the ratio of the numerator and the denominator polynomials, i.e., (G(s)=frac).

Definition: Transfer Function Zeros

The roots of the numerator polynomial, (n(s)), define system zeros, i.e., those frequencies at which the system response is zero. Thus, (z_0) is a zero of the transfer function if (Gleft(z_0 ight)=0.)

Definition: Transfer Function Poles

The roots of the denominator polynomial, (d(s)), define system poles, i.e., those frequencies at which the system response is infinite. Thus, (p_0) is a pole of the transfer function if (Gleft(p_0 ight)=infty .)

The poles and zeros of first and second-order system models are described below.

First-Order System

A first-order system has a generic ODE description: ( au dotleft(t ight)+yleft(t ight)=u(t)), where (uleft(t ight)) and (yleft(t ight)) denote the input and the output, and ( au) is the system time constant. By applying the Laplace transform, a first-order transfer function is obtained as: [G(s)=frac< au s+1>]

The transfer function has no finite zeros and a single pole located at (s=-frac<1>< au >) in the complex plane.

The reduced-order model of a DC motor with voltage input and angular velocity output (Example 1.4.3) is described by the differential equation: ( au dotomega (t) + omega(t) = V_a(t)).

The DC motor has a transfer function: (G(s)=frac< au_m s+1>) where ( au_m) is the motor time constant.

For the following parameter values: (R=1Omega , L=0.01H, J=0.01 kgm^ <2>, b=0.1 frac , and k_ =k_ =0.05), the motor transfer function evaluates as:

The transfer function has a single pole located at: (s=-10.25) with associated time constant of (0.098 sec).

Second-Order System with an Integrator

A first-order system with an integrator is described by the transfer function:

The system has no finite zeros and has two poles located at (s=0) and (s=-frac<1>< au >) in the complex plane.

The DC motor modeled in Example 2.1.1 above is used in a position control system where the objective is to maintain a certain shaft angle ( heta(t)). The motor equation is given as: ( au ddot heta(t) + dot heta(t) = V_a(t)) its transfer function is given as: (Gleft(s ight)=frac).

Using the above parameter values in the reduced-order DC motor model, the system transfer function is given as:

The transfer function has no finite zeros and poles are located at: (s=0,-10.25).

Second-Order System with Real Poles

A second-order system with poles located at (s=-_1, -_2) is described by the transfer function:

From Section 1.4, the DC motor transfer function is described as: [G(s)=frac <(s+1/ au _)(s+1/ au _ )>]

Then, system poles are located at: (s_ <1>=-frac<1> < au _>) and (s_ <2>=-frac<1> < au _>), where ( au_e) and ( au_) represent the electrical and mechanical time constants of the motor.

For the following parameter values: (R=1Omega , L=0.01H, J=0.01 kgm^ <2>, b=0.1 frac , and k_ =k_ =0.05), the transfer function from armature voltage to angular velocity is given as:

The transfer function poles are located at: (s=-10.28, -99.72).

The motor time constants are given as: ( au _ cong frac=10 ms, au _ m cong frac=100 ms).

Second-Order System with Complex Poles

A second-order model with its complex poles located at: (s=-sigma pm jomega) is described by the transfer function:

Equivalently, the second-order transfer function with complex poles is expressed in terms of the damping ratio, (zeta), and the natural frequency, (_n), of the complex poles as:

The transfer function poles are located at: (s_<1,2>=-zeta _npm j_d), where (_d=omega_nsqrt<1-^2>) (Figure 2.1.1).

As seen from the figure, (_n) equals the magnitude of the complex pole, and (zeta =frac<_n>=), where ( heta) is the angle subtended by the complex pole at the origin.

The damping ratio, (zeta), is a dimensionless quantity that characterizes the decay of the oscillations in the system&rsquos natural response. The damping ratio is bounded as: (0<zeta <1).

1. As (zeta o 0), the complex poles are located close to the imaginary axis at: (scong pm j_n). The resulting impulse response displays persistent oscillations at system&rsquos natural frequency, (_n).
2. As (zeta o 1), the complex poles are located close to the real axis as (s_<1,2>cong -zeta _n). The resulting impulse response has no oscillations and exponentially decays to zero resembling the response of a first-order system.

A spring&ndashmass&ndashdamper system has a transfer function:

Its characteristic equation is given as: (ms^s+bs+k=0), whose roots are characterized by the sign of the discriminant, (Delta =b^ <2>-4mk).

For (Delta >0,) the system has real poles, located at:

Next, assume that the mass-spring-damper has the following parameter values: (m=1, b=k=2) then, its transfer function is given as: [G(s)=frac<1>=frac<1>]

The transfer function has complex poles located at: (s=-1pm j1).

Further, the complex poles have an angle: ( heta=45^circ), and (cos45^circ=frac<1>>).

Figure (PageIndex<1>): Second-order transfer function pole locations in the complex plane.

Different Is Good: FreshDirect Redefines the NYC Grocery Landscape

For an example of the relationship between technology and strategic positioning, consider FreshDirect. The New York City–based grocery firm focused on the two most pressing problems for Big Apple shoppers: selection is limited and prices are high. Both of these problems are a function of the high cost of real estate in New York. The solution? Use technology to craft an ultraefficient model that makes an end-run around stores.

The firm’s “storefront” is a Web site offering one-click menus, semiprepared specials like “meals in four minutes,” and the ability to pull up prior grocery lists for fast reorders—all features that appeal to the time-strapped Manhattanites who were the firm’s first customers. (The Web’s not the only channel to reach customers—the firm’s iPhone app was responsible for 2.5 percent of sales just weeks after launch)(Schneiderman, 2010). Next-day deliveries are from a vast warehouse the size of five football fields located in a lower-rent industrial area of Queens. At that size, the firm can offer a fresh goods selection that’s over five times larger than local supermarkets. Area shoppers—many of whom don’t have cars or are keen to avoid the traffic-snarled streets of the city—were quick to embrace the model. The service is now so popular that apartment buildings in New York have begun to redesign common areas to include secure freezers that can accept FreshDirect deliveries, even when customers aren’t there (Croghan, 2006).

Figure 2.1 The FreshDirect Web Site and the Firm’s Tech-Enabled Warehouse Operation

The FreshDirect model crushes costs that plague traditional grocers. Worker shifts are highly efficient, avoiding the downtime lulls and busy rush hour spikes of storefronts. The result? Labor costs that are 60 percent lower than at traditional grocers. FreshDirect buys and prepares what it sells, leading to less waste, an advantage that the firm claims is “worth 5 percentage points of total revenue in terms of savings” (Fox, 2009). Overall perishable inventory at FreshDirect turns 197 times a year versus 40 times a year at traditional grocers (Schonfeld, 2004). Higher inventory turns mean the firm is selling product faster, so it collects money quicker than its rivals do. And those goods are fresher since they’ve been in stock for less time, too. Consider that while the average grocer may have seven to nine days of seafood inventory, FreshDirect’s seafood stock turns each day. Stock is typically purchased direct from the docks in order to fulfill orders placed less than twenty-four hours earlier (Laseter, et. al., 2003).

Artificial intelligence software, coupled with some seven miles of fiber-optic cables linking systems and sensors, supports everything from baking the perfect baguette to verifying orders with 99.9 percent accuracy (Black, 2002 Sieber & Mitchell, 2002). Since it lacks the money-sucking open-air refrigerators of the competition, the firm even saves big on energy (instead, staff bundle up for shifts in climate-controlled cold rooms tailored to the specific needs of dairy, deli, and produce). And a new initiative uses recycled biodiesel fuel to cut down on delivery costs.

FreshDirect buys directly from suppliers, eliminating middlemen wherever possible. The firm also offers suppliers several benefits beyond traditional grocers, all in exchange for more favorable terms. These include offering to carry a greater selection of supplier products while eliminating the “slotting fees” (payments by suppliers for prime shelf space) common in traditional retail, cobranding products to help establish and strengthen supplier brand, paying partners in days rather than weeks, and sharing data to help improve supplier sales and operations. Add all these advantages together and the firm’s big, fresh selection is offered at prices that can undercut the competition by as much as 35 percent (Green, 2003). And FreshDirect does it all with margins in the range of 20 percent (to as high as 45 percent on many semiprepared meals), easily dwarfing the razor-thin 1 percent margins earned by traditional grocers.

Today, FreshDirect serves a base of some 600,000 paying customers. That’s a population roughly the size of metro-Boston, serviced by a single grocer with no physical store. The privately held firm has been solidly profitable for several years. Even in recession-plagued 2009, the firm’s CEO described 2009 earnings as “pretty spectacular,” while 2010 revenues are estimated to grow to roughly $300 million (Schneiderman, 2010). Technology is critical to the FreshDirect model, but it’s the collective impact of the firm’s differences when compared to rivals, this tech-enabled strategic positioning, that delivers success. Operating for more than half a decade, the firm has also built up a set of strategic assets that not only address specific needs of a market but are now extremely difficult for any upstart to compete against. Traditional grocers can’t fully copy the firm’s delivery business because this would leave them straddling two markets (low-margin storefront and high-margin delivery), unable to gain optimal benefits from either. Entry costs for would-be competitors are also high (the firm spent over$75 million building infrastructure before it could serve a single customer), and the firm’s complex and highly customized software, which handles everything from delivery scheduling to orchestrating the preparation of thousands of recipes, continues to be refined and improved each year (Valerio, 2009). On top of all this comes years of customer data used to further refine processes, speed reorders, and make helpful recommendations. Competing against a firm with such a strong and tough-to-match strategic position can be brutal. Just five years after launch there were one-third fewer supermarkets in New York City than when FreshDirect first opened for business (Shulman, 2008).

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