A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

The domain is [latex]\left\{1,2,3,4,5\right\}[/latex]. The range is [latex]\left\{2,4,6,8,10\right\}[/latex].

Note that each value in the domain is also known as an input value. The input values are values of the independent variable which often labeled with the lowercase letter [latex]x[/latex]. Each value in the range is also known as an output value. The output values are values of the dependent variable which is often labeled lowercase letter [latex]y[/latex].

A function [latex]f[/latex] is a relation that assigns a single value in the range to each value in the domain. In other words no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, [latex]\left\{1,2,3,4,5\right\}[/latex], is paired with exactly one element in the range, [latex]\left\{2,4,6,8,10\right\}[/latex].

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain, [latex]\left\{\text{even,}\text{odd}\right\}[/latex] is not paired with exactly one element in the range, [latex]\left\{1,2,3,4,5\right\}[/latex]. For example, the term “odd” corresponds to three values from the domain, [latex]\left\{1,3,5\right\}[/latex] and the term “even” corresponds to two values from the range, [latex]\left\{2,4\right\}[/latex]. This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

Figure 1. (a) This relationship is a function because each input is associated with a single output. Note that input [latex]q[/latex] and [latex]r[/latex] both give output [latex]n[/latex]. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input [latex]q[/latex] is associated with two different outputs.

Example 1: Determining If Menu Price Lists Are Functions

The coffee shop menu, shown in Figure 2 consists of items and their prices.

1. Is price a function of the item?
2. Is the item a function of the price?

 

Figure 2

Show Solution

1. Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure 2.
   

   

   Figure 2

   Each item on the menu has only one price, so the price is a function of the item.

2. Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See Figure 3.
   

   

   Figure 3

   Therefore, the item is a not a function of price.

Example 2: Determining If Class Grade Rules Are Functions

In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.

Percent Grade	0–56	57–61	62–66	67–71	72–77	78–86	87–91	92–100
Grade Point Average	0.0	1.0	1.5	2.0	2.5	3.0	3.5	4.0

Show Solution

For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.

In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

Try It

The table below lists the five greatest baseball players of all time in order of rank.

Player	Rank
Babe Ruth	1
Willie Mays	2
Ty Cobb	3
Walter Johnson	4
Hank Aaron	5

a) Is the rank a function of the player name?

b) Is the player name a function of the rank?

Show Solution

a. yes;

b. yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables [latex]h[/latex] for height and [latex]a[/latex] for age. The letters [latex]f,g[/latex], and [latex]h[/latex] are often used to represent functions just as we use [latex]x,y[/latex], and [latex]z[/latex] to represent numbers and [latex]A,B[/latex], and [latex]C[/latex] to represent sets.

[latex]\begin{gathered}\begin{cases}\begin{align}&h\text{ is }f\text{ of }a && \text{We name the function }f;\text{ height is a function of age}. \\ &h=f\left(a\right) && \text{We use parentheses to indicate the function input}\text{. } \\ &f\left(a\right) && \text{We name the function }f;\text{ the expression is read as ``}f\text{ of }a\text{.''} \end{align} \end{cases}\end{gathered}[/latex]

Remember, we can use any letter to name the function; the notation [latex]h\left(a\right)[/latex] shows us that [latex]h[/latex] depends on [latex]a[/latex]. The value [latex]a[/latex] must be put into the function [latex]h[/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example [latex]f\left(a+b\right)[/latex] means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

A General Note: Function Notation

The notation [latex]y=f\left(x\right)[/latex] defines a function named [latex]f[/latex]. This is read as [latex]``y[/latex] is a function of [latex]x.''[/latex] The letter [latex]x[/latex] represents the input value, or independent variable. The letter [latex]y[/latex], or [latex]f\left(x\right)[/latex], represents the output value, or dependent variable.

Example 3: Using Function Notation for Days in a Month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.

Show Solution

The number of days in a month is a function of the name of the month, so if we name the function [latex]f[/latex], we write [latex]\text{days}=f\left(\text{month}\right)[/latex] or [latex]d=f\left(m\right)[/latex]. The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

Figure 4

For example, [latex]f\left(\text{March}\right)=31[/latex], because March has 31 days. The notation [latex]d=f\left(m\right)[/latex] reminds us that the number of days, [latex]d[/latex] (the output), is dependent on the name of the month, [latex]m[/latex] (the input).

Analysis of the Solution

Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

Example 4: Interpreting Function Notation

A function [latex]N=f\left(y\right)[/latex] gives the number of police officers, [latex]N[/latex], in a town in year [latex]y[/latex]. What does [latex]f\left(2005\right)=300[/latex] represent?

Show Solution

When we read [latex]f\left(2005\right)=300[/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]\left(N\right)[/latex], is 300. Remember, [latex]N=f\left(y\right)[/latex]. The statement [latex]f\left(2005\right)=300[/latex] tells us that in the year 2005 there were 300 police officers in the town.

Q & A

Instead of a notation such as [latex]y=f\left(x\right)[/latex], could we use the same symbol for the output as for the function, such as [latex]y=y\left(x\right)[/latex], meaning “y is a function of x?”

Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as [latex]f[/latex], which is a rule or procedure, and the output [latex]y[/latex] we get by applying [latex]f[/latex] to a particular input [latex]x[/latex]. This is why we usually use notation such as [latex]y=f\left(x\right),P=W\left(d\right)[/latex], and so on.

Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

The table below lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[/latex] where [latex]D=f\left(m\right)[/latex] identifies months by an integer rather than by name.

Month number, [latex]m[/latex] (input)	1	2	3	4	5	6	7	8	9	10	11	12
Days in month, [latex]D[/latex] (output)	31	28	31	30	31	30	31	31	30	31	30	31

The table below defines a function [latex]Q=g\left(n\right)[/latex]. Remember, this notation tells us that [latex]g[/latex] is the name of the function that takes the input [latex]n[/latex] and gives the output [latex]Q[/latex].

[latex]n[/latex]	1	2	3	4	5
[latex]Q[/latex]	8	6	7	6	8

The table below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

Age in years, [latex]\text{ }a\text{ }[/latex] (input)	5	5	6	7	8	9	10
Height in inches, [latex]\text{ }h\text{ }[/latex] (output)	40	42	44	47	50	52	54

How To: Given a table of input and output values, determine whether the table represents a function.

1. Identify the input and output values.
2. Check to see if each input value is paired with only one output value. If so, the table represents a function.

Example 5: Identifying Tables that Represent Functions

Which table, a), b), or c), represents a function (if any)?

a)

Table A

Input	Output
2	1
5	3
8	6

b)

Table B

Input	Output
–3	5
0	1
4	5

c)

Table C

Input	Output
1	0
5	2
5	4

Show Solution

a) and b) define functions. In both, each input value corresponds to exactly one output value. c) does not define a function because the input value of 5 corresponds to two different output values.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by a) can be represented by writing

[latex]f\left(2\right)=1,f\left(5\right)=3,\text{and }f\left(8\right)=6[/latex]

Similarly, the statements [latex]g\left(-3\right)=5,g\left(0\right)=1,\text{and }g\left(4\right)=5[/latex] represent the function in b).

c) cannot be expressed in a similar way because it does not represent a function.

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluating Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example the function [latex]f\left(x\right)=5 - 3{x}^{2}[/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

How To: Given the formula for a function, evaluate.

1. Replace the input variable in the formula with the value provided.
2. Calculate the result.

Example 6: Evaluating Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], evaluate [latex]h\left(4\right)[/latex].

Show Solution

To evaluate [latex]h\left(4\right)[/latex], we substitute the value 4 for the input variable [latex]p[/latex] in the given function.

[latex]\begin{align}h\left(p\right)&={p}^{2}+2p \\ h\left(4\right)&={\left(4\right)}^{2}+2\left(4\right) \\ &=16+8 \\ &=24 \end{align}[/latex]

Therefore, for an input of 4, we have an output of 24.

Example 7: Evaluating Functions at Specific Values

Evaluate [latex]f\left(x\right)={x}^{2}+3x - 4[/latex] at

1. [latex]2[/latex]
2. [latex]a[/latex]
3. [latex]a+h[/latex]
4. [latex]\frac{f\left(a+h\right)-f\left(a\right)}{h}[/latex]

Show Solution

Replace the [latex]x[/latex] in the function with each specified value.

1. Because the input value is a number, 2, we can use algebra to simplify.
   [latex]\begin{align}f\left(2\right)&={2}^{2}+3\left(2\right)-4 \\ &=4+6 - 4 \\ &=6 \end{align}[/latex]
2. In this case, the input value is a letter so we cannot simplify the answer any further.
   [latex]f\left(a\right)={a}^{2}+3a - 4[/latex]
3. With an input value of [latex]a+h[/latex], we must use the distributive property.
   [latex]\begin{align}f\left(a+h\right)&={\left(a+h\right)}^{2}+3\left(a+h\right)-4 \\ &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \end{align}[/latex]
4. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that
   [latex]f\left(a+h\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[/latex]

   and we know that

   [latex]f\left(a\right)={a}^{2}+3a - 4[/latex]

   Now we combine the results and simplify.

   [latex]\begin{align} \frac{f\left(a+h\right)-f\left(a\right)}{h}&=\frac{\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\right)-\left({a}^{2}+3a - 4\right)}{h} \\[1.5mm]&=\frac{2ah+{h}^{2}+3h}{h} \\[1.5mm]&=\frac{h\left(2a+h+3\right)}{h} &&\text{Factor out }h. \\[1.5mm]&=2a+h+3 && \text{Simplify}. \end{align}[/latex]

Try It

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], evaluate [latex]g\left(5\right)[/latex].

Show Solution

[latex]g\left(5\right)=1[/latex]

Example 8: Solving Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], solve for [latex]h\left(p\right)=3[/latex].

Show Solution

[latex]\begin{align}h\left(p\right)=3 \\ {p}^{2}+2p=3 &\hspace{3mm} \text{Substitute the original function }h\left(p\right)={p}^{2}+2p. \\ {p}^{2}+2p - 3=0 &\hspace{3mm} \text{Subtract 3 from each side}. \\ \left(p+3\text{)(}p - 1\right)=0 &\hspace{3mm} \text{Factor}. \end{align}[/latex]

If [latex]\left(p+3\right)\left(p - 1\right)=0[/latex], either [latex]\left(p+3\right)=0[/latex] or [latex]\left(p - 1\right)=0[/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[/latex] in each case.

[latex]\begin{align}\left(p+3\right)=0, & \hspace{3mm} p=-3 \\ \left(p - 1\right)=0, & \hspace{3mm} p=1 \end{align}[/latex]

This gives us two solutions. The output [latex]h\left(p\right)=3[/latex] when the input is either [latex]p=1[/latex] or [latex]p=-3[/latex].

Figure 5

We can also verify by graphing as in Figure 5. The graph verifies that [latex]h\left(1\right)=h\left(-3\right)=3[/latex] and [latex]h\left(4\right)=24[/latex].

Try It

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], solve [latex]g\left(m\right)=2[/latex].

Show Solution

[latex]m=8[/latex]

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[/latex] expresses a functional relationship between [latex]n[/latex] and [latex]p[/latex]. We can rewrite it to decide if [latex]p[/latex] is a function of [latex]n[/latex].

How To: Given a function in equation form, write its algebraic formula.

1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example 9: Finding an Equation of a Function

Express the relationship [latex]2n+6p=12[/latex] as a function [latex]p=f\left(n\right)[/latex], if possible.

Show Solution

To express the relationship in this form, we need to be able to write the relationship where [latex]p[/latex] is a function of [latex]n[/latex], which means writing it as p = expression involving n.

[latex]\begin{align} &2n+6p=12 \\ &6p=12 - 2n && \text{Subtract }2n\text{ from both sides}. \\ &p=\frac{12 - 2n}{6} && \text{Divide both sides by 6 and simplify}. \\ &p=\frac{12}{6}-\frac{2n}{6} \\ &p=2-\frac{1}{3}n \end{align}[/latex]

Therefore, [latex]p[/latex] as a function of [latex]n[/latex] is written as

[latex]p=f\left(n\right)=2-\frac{1}{3}n[/latex]

Analysis of the Solution

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Example 10: Expressing the Equation of a Circle as a Function

Does the equation [latex]{x}^{2}+{y}^{2}=1[/latex] represent a function with [latex]x[/latex] as input and [latex]y[/latex] as output? If so, express the relationship as a function [latex]y=f\left(x\right)[/latex].

Show Solution

First we subtract [latex]{x}^{2}[/latex] from both sides.

[latex]\begin{align} &{x}^{2}+{y}^{2}=1 \\ &{y}^{2}=1-{x}^{2} && \text{Subtract } {x}^{2} \text{ from both sides} \\ &y=\pm \sqrt{1-{x}^{2}} && \text{Solve for } y \text{ using the square root principle.} \end{align}[/latex]

 

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function

[latex]y=f\left(x\right)[/latex]

Try It

If [latex]x - 8{y}^{3}=0[/latex], express [latex]y[/latex] as a function of [latex]x[/latex].

Show Solution

[latex]y=f\left(x\right)=\frac{\sqrt[3]{x}}{2}[/latex]

Q & A

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y}[/latex], if we want to express [latex]y[/latex] as a function of [latex]x[/latex], there is no simple algebraic formula involving only [latex]x[/latex] that equals [latex]y[/latex]. However, each [latex]x[/latex] does determine a unique value for [latex]y[/latex], and there are mathematical procedures by which [latex]y[/latex] can be found to any desired accuracy. In this case we say that the equation gives an implicit (implied) rule for [latex]y[/latex] as a function of [latex]x[/latex], even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.

Pet	Memory span in hours
Puppy	0.008
Adult dog	0.083
Cat	16
Goldfish	2160
Beta fish	3600

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[/latex].

The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function [latex]P[/latex] at the input value of “goldfish.” We would write [latex]P\left(\text{goldfish}\right)=2160[/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.

How To: Given a function represented by a table, identify specific output and input values.

1. Find the given input in the row (or column) of input values.
2. Identify the corresponding output value paired with that input value.
3. Find the given output values in the row (or column) of output values, noting every time that output value appears.
4. Identify the input value(s) corresponding to the given output value.

Example 11: Evaluating and Solving a Tabular Function

Using the table below,

1. Evaluate [latex]g\left(3\right)[/latex].
2. Solve [latex]g\left(n\right)=6[/latex].

n	1	2	3	4	5
g(n)	8	6	7	6	8

Show Solution

• Evaluating [latex]g\left(3\right)[/latex] means determining the output value of the function [latex]g[/latex] for the input value of [latex]n=3[/latex]. The table output value corresponding to [latex]n=3[/latex] is 7, so [latex]g\left(3\right)=7[/latex].
• Solving [latex]g\left(n\right)=6[/latex] means identifying the input values, [latex]n[/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[/latex] and [latex]n=4[/latex].

n	1	2	3	4	5
g(n)	8	6	7	6	8

When we input 2 into the function [latex]g[/latex], our output is 6. When we input 4 into the function [latex]g[/latex], our output is also 6.

Try It

Using the table in Example 11, evaluate [latex]g\left(1\right)[/latex] .

Show Solution

[latex]g\left(1\right)=8[/latex]

Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value [latex]x[/latex] and the output value [latex]y[/latex], and we say [latex]y[/latex] is a function of [latex]x[/latex], or [latex]y=f\left(x\right)[/latex] when the function is named [latex]f[/latex]. The graph of the function is the set of all points [latex]\left(x,y\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right)[/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 6 tell us that [latex]f\left(0\right)=2[/latex] and [latex]f\left(6\right)=1[/latex]. However, the set of all points [latex]\left(x,y\right)[/latex] satisfying [latex]y=f\left(x\right)[/latex] is a curve. The curve shown includes [latex]\left(0,2\right)[/latex] and [latex]\left(6,1\right)[/latex] because the curve passes through those points.

Figure 6

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.

Figure 7

How To: Given a graph, use the vertical line test to determine if the graph represents a function.

1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
2. If there is any such line, then the graph does not represent a function.

Example 12: Applying the Vertical Line Test

Which of the graphs represent(s) a function [latex]y=f\left(x\right)?[/latex]

Figure 8

Show Solution

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 8. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point.

Figure 9

Try It

Does the graph in Figure 10 represent a function?

Figure 10

Show Solution

Yes.

Find the domain of a function defined by an equation

In this section we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

Figure 11

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products.

We can write the domain and range in interval notation, which uses values within brackets or parentheses to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write [latex]\left(0,\text{ }100\right][/latex]. We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or even root, the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, exclude values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The lowest term from the interval is written first.
• The greatest term in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

The table below gives a summary of interval notation.

Example 13: Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function: [latex]\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(4,\text{ }20\right),\left(5,\text{ }30\right),\left(6,\text{ }40\right)\right\}[/latex] .

Show Solution

First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

[latex]\left\{2,3,4,5,6\right\}[/latex]

Try It

Find the domain of the function:

[latex]\left\{\left(-5,4\right),\left(0,0\right),\left(5,-4\right),\left(10,-8\right),\left(15,-12\right)\right\}[/latex]

Show Solution

[latex]\left\{-5,0,5,10,15\right\}[/latex]

How To: Given a function written in equation form, find the domain.

1. Identify the input values.
2. Identify any restrictions on the input and exclude those values from the domain.
3. Write the domain in interval form, if possible.

Example 14: Finding the Domain of a Function

Find the domain of the function [latex]f\left(x\right)={x}^{2}-1[/latex].

Show Solution

The input value, shown by the variable [latex]x[/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].

Try It

Find the domain of the function: [latex]f\left(x\right)=5-x+{x}^{3}[/latex].

Show Solution

[latex]\left(-\infty ,\infty \right)[/latex]

Try It

How To: Given a function written in an equation form that includes a fraction, find the domain.

1. Identify the input values.
2. Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for [latex]x[/latex] . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
3. Write the domain in interval form, making sure to exclude any restricted values from the domain.

Example 15: Finding the Domain of a Function Involving a Denominator (Rational Function)

Find the domain of the function [latex]f\left(x\right)=\frac{x+1}{2-x}[/latex].

Show Solution

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[/latex].

[latex]\begin{gathered}2-x=0 \\ -x=-2 \\ x=2 \\ \text{ } \end{gathered}[/latex]

Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[/latex] or [latex]x>2[/latex]. We can use a symbol known as the union, [latex]\cup [/latex], to combine the two sets. In interval notation, we write the solution: [latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right)[/latex].

Figure 12

In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,2\right)\cup \left(2,\infty \right)[/latex].

Try It

Find the domain of the function: [latex]f\left(x\right)=\frac{1+4x}{2x - 1}[/latex].

Show Solution

[latex]\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)[/latex]

Try It

How To: Given a function written in equation form including an even root, find the domain.

1. Identify the input values.
2. Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[/latex].
3. The solution(s) are the domain of the function. If possible, write the answer in interval form.

Example 16: Finding the Domain of a Function with an Even Root

Find the domain of the function [latex]f\left(x\right)=\sqrt{7-x}[/latex].

Show Solution

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for [latex]x[/latex].

[latex]\begin{gathered}7-x\ge 0 \\ -x\ge -7 \\ x\le 7\\ \text{ } \end{gathered}[/latex]

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[/latex], or [latex]\left(-\infty ,7\right][/latex].

Try It

Find the domain of the function [latex]f\left(x\right)=\sqrt{5+2x}[/latex].

Show Solution

[latex]\left[-\frac{5}{2},\infty \right)[/latex]

Try It

Q & A

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function [latex]f\left(x\right)=-\frac{1}{\sqrt{x}}[/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

The Difference Quotient

Let’s imagine a point on the curve of function [latex]f[/latex] at x as shown in Figure 13. The coordinates of the point are [latex]\left(x,f\left(x\right)\right)[/latex]. Connect this point with a second point on the curve a little to the right of [latex]x[/latex], with an x-value increased by some small real number [latex]h[/latex]. The coordinates of this second point are [latex]\left(x+h,f\left(x+h\right)\right)[/latex] for some positive-value [latex]h[/latex].

Figure 13. Connecting point [latex]a[/latex] with a point just beyond allows us to measure a slope close to that of a tangent line at x.

We can calculate the slope of the line connecting the two points [latex]\left(x,f\left(x\right)\right)[/latex] and [latex]\left(x+h,f\left(x+h\right)\right)[/latex], called a secant line, by applying the slope formula,

[latex]\text{slope = }\dfrac{\text{change in }y}{\text{change in }x}[/latex]

[latex]\text{slope = }\dfrac{\text{change in }y}{\text{change in }x}[/latex]

We use the notation [latex]{m}_{\sec }[/latex] to represent the slope of the secant line connecting two points.

[latex]\begin{align}{m}_{\sec }&=\dfrac{f\left(x+h\right)-f\left(x\right)}{\left(x+h\right)-\left(x\right)} \\ &=\dfrac{f\left(x+h\right)-f\left(x\right)}{h} \end{align}[/latex]

The slope [latex]{m}_{\sec }[/latex] equals the difference quotient between two points [latex]\left(x,f\left(x\right)\right)[/latex] and [latex]\left(x+h,f\left(x+h\right)\right)[/latex].

THE DIFFERENCE QUOTIENT

The difference quotient between two points [latex]\left(x,f\left(x\right)\right)[/latex] and [latex]\left(x+h,f\left(x+h\right)\right)[/latex] on the curve of [latex]f[/latex] is the slope of the line connecting the two points and is given by

[latex]\text{difference quotient}=\dfrac{f\left(x+h\right)-f\left(x\right)}{h}[/latex]

Example 17: Finding the difference quotient (Quadratic function)

Find the difference quotient if [latex]f(x)=2x^{2}-x[/latex].

Show Solution

We use the difference quotient formula.

​[latex]\begin{align}\frac{f\left(x+h\right)-f\left(x\right)}{h}\hspace{.5in} & \text{Start with the difference quotient} \\ \frac{2(x+h)^{2}-(x+h)-(2x^{2}-x)}{h}\hspace{.5in} & \text{Replace x with (x+h)} \\ \frac{2(x^{2}+2xh+h^{2})-x-h-2x^{2}+x}{h}\hspace{.5in} & \text{Distribute} \\ \frac{2x^2+4xh+2h^2-x-h-2x^2+x}{h}\hspace{.5in} & \text{Distribute} \\ \frac{4xh+2h^{2}-h}{h}\hspace{.5in} & \text{Simplify like terms} \\ 4x+2h-1\hspace{.5in} & \text{This is the difference quotient}\end{align}[/latex]

Try It

Find the difference quotient if [latex]f\left(x\right)={x}^{2}+2x - 8[/latex].

Show Solution

[latex]2x+h+2[/latex]

Example 18: Finding the difference quotient (rational function)

Find the difference quotient if [latex]f(x)=\dfrac{1}{x}[/latex].

Show Solution

We use the difference quotient formula.

​[latex]\begin{align}\dfrac{f\left(x+h\right)-f\left(x\right)}{h}\hspace{.5in} & \text{Start with the difference quotient} \\ \dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}\hspace{.5in} & \text{Replace x with (x+h)} \\ \dfrac{\dfrac{x}{x(x+h)}-\dfrac{x+h}{x(x+h)}}{h}\hspace{.5in} & \text{Get common denominators} \\ \dfrac{\dfrac{x-x-h}{x(x+h)}}{h}\hspace{.5in} & \text{Subtract the fractions} \\ \dfrac{-h}{x(x+h)h}\hspace{.5in} & \text{Flip and multiply} \\ \dfrac{-1}{x(x+h)}\hspace{.5in} & \text{This is the difference quotient}\end{align}[/latex]

Try It

Find the difference quotient if [latex]f(x)=-\dfrac{2}{x}[/latex].

Show Solution

[latex]\dfrac{2}{x(x+h)}[/latex]

Key Concepts

• • • A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.
    • Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\left(x\right)[/latex].
    • In tabular form, a function can be represented by rows or columns that relate to input and output values.
    • To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
    • To solve for a specific function value, we determine the input values that yield the specific output value.
    • An algebraic form of a function can be written from an equation.
    • Input and output values of a function can be identified from a table.
    • Relating input values to output values on a graph is another way to evaluate a function.
    • A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.
    • The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
    • The domain of a function can be determined by listing the input values of a set of ordered pairs.
    • The domain of a function can also be determined by identifying the input values of a function written as an equation.
    • The difference quotient is used to calculate the slope of the secant line between two points on the graph of a function, f.

Glossary

dependent variable

an output variable

domain

the set of all possible input values for a relation

function

a relation in which each input value yields a unique output value

independent variable

an input variable

input

each object or value in a domain that relates to another object or value by a relationship known as a function

output

each object or value in the range that is produced when an input value is entered into a function

range

the set of output values that result from the input values in a relation

relation

a set of ordered pairs

vertical line test

a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once

 

Section 2.1 Homework Exercises

1. What is the difference between a relation and a function?

2. What is the difference between the input and the output of a function?

3. Why does the vertical line test tell us whether the graph of a relation represents a function?

For the following exercises, determine whether the relation represents a function.

4. [latex]\{(a,b),(c,d),(a,c)\}[/latex]

5. [latex]\{(a,b),(b,c),(c,c)\}[/latex]

For the following exercises, determine whether the relation represents [latex]y[/latex] as a function of [latex]x[/latex].

6. [latex]5x+2y=10[/latex]

7. [latex]y=x^{2}[/latex]

8. [latex]x=y^{2}[/latex]

9. [latex]3x^{2}+y=14[/latex]

10. [latex]2x+y^{2}=6[/latex]

11. [latex]y=−2x^{2}+40x[/latex]

12. [latex]y=\frac{1}{x}[/latex]

13. [latex]x=\frac{3y+5}{7y−1}[/latex]

14. [latex]x=\sqrt{1−y^2}[/latex]

15. [latex]y=\frac{3x+5}{7x−1}[/latex]

16. [latex]x^2+y^2=9[/latex]

17. [latex]2xy=1[/latex]

18. [latex]x=y^3[/latex]

19. [latex]y=x^3[/latex]

20. [latex]y=\sqrt{1−x^2}[/latex]

21. [latex]x=\pm\sqrt{1−y}[/latex]

22. [latex]y=\pm\sqrt{1−x}[/latex]

23. [latex]y^2=x^2[/latex]

24. [latex]y^3=x^2[/latex]

For the following exercises, evaluate the function [latex]f[/latex] at the indicated values [latex]f(−3),f(2),f(−a),−f(a),f(a+h)[/latex].

25. [latex]f(x)=2x−5[/latex]

26. [latex]f(x)=−5x^2+2x−1[/latex]

27. [latex]f(x)=\sqrt{2−x}+5[/latex]

28. [latex]f(x)=\frac{6x−15}{x+2}[/latex]

29. [latex]f(x)=|x−1|−|x+1|[/latex]

30. Given the function [latex]g(x)=5−x^{2}[/latex],evaluate [latex]g(x+h)−g(x)h,h\ne{0}[/latex].

31. Given the function [latex]g(x)=x^{2}+2x[/latex],evaluate [latex]g(x)−g(a)x−a,x\ne{a}[/latex].

32. Given the function [latex]k(t)=2t−1[/latex]:

Evaluate [latex]k(2)[/latex].

Solve [latex]k(t)=7[/latex].

33. Given the function [latex]f(x)=8−3x[/latex]:
Evaluate [latex]f(−2)[/latex].
Solve [latex]f(x)=−1[/latex].

34. Given the function [latex]p(c)=c^2+c[/latex]:
Evaluate [latex]p(−3)[/latex].
Solve [latex]p(c)=2[/latex].

35. Given the function [latex]f(x)=x^2−3x[/latex]:
Evaluate [latex]f(5)[/latex].
Solve [latex]f(x)=4[/latex].

36. Given the function [latex]f(x)=\sqrt{x+2}[/latex]:
Evaluate [latex]f(7)[/latex].
Solve [latex]f(x)=4[/latex].

37. Consider the relationship [latex]3r+2t=18[/latex].
Write the relationship as a function [latex]r=f(t)[/latex].
Evaluate [latex]f(−3)[/latex].
Solve [latex]f(t)=2[/latex].

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50. Given the following graph,
Evaluate [latex]f(−1)[/latex].
Solve for [latex]f(x)=3[/latex].

51. Given the following graph,
Evaluate [latex]f(0[/latex]).
Solve for [latex]f(x)=−3[/latex].

52. Given the following graph,
Evaluate [latex]f(4)[/latex].
Solve for [latex]f(x)=1[/latex].

For the following exercises, determine whether the relation represents a function.

53. [latex]\{(−1,−1),(−2,−2),(−3,−3)\}[/latex]

54. [latex]\{(3,4),(4,5),(5,6)\}[/latex]

55. [latex]\{(2,5),(7,11),(15,8),(7,9)\}[/latex]

For the following exercises, determine if the relation represented in table form represents [latex]y[/latex] as a function of [latex]x[/latex].

56.

57.

58.
For the following exercises, use the function [latex]f[/latex] represented in the table below.

59. Evaluate [latex]f(3)[/latex].

60. Solve [latex]f(x)=1[/latex].

For the following exercises, evaluate the function [latex]f[/latex] at the values [latex]f(−2),f(−1),f(0),f(1),\text{ and }f(2)[/latex].

61. [latex]f(x)=4−2x[/latex]

62. [latex]f(x)=8−3x[/latex]

63. [latex]f(x)=8x^2−7x+3[/latex]

64. [latex]f(x)=3+\sqrt{x+3}[/latex]

65. [latex]f(x)=\frac{x-2}{x+3}[/latex]

66. [latex]3^x[/latex]

For the following exercises, evaluate the expressions, given functions [latex]f,g,\text{ and }h[/latex]:

[latex]f(x)=3x−2[/latex]
[latex]g(x)=5−x^2[/latex]
[latex]h(x)=−2x^2+3x−1[/latex]

67. [latex]3f(1)−4g(−2)[/latex]

68. [latex]f(73)−h(−2)[/latex]

69. The amount of garbage, [latex]G[/latex], produced by a city with population [latex]p[/latex] is given by [latex]G=f(p)[/latex]. [latex]G[/latex] is measured in tons per week, and [latex]p[/latex] is measured in thousands of people. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function [latex]f[/latex]. Explain the meaning of the statement [latex]f(5)=2[/latex].

70. The number of cubic yards of dirt, [latex]D[/latex],needed to cover a garden with area a square feet is given by [latex]D=g(a)[/latex]. A garden with area 5000 ft² requires 50 yd³ of dirt. Express this information in terms of the function [latex]g[/latex].  Explain the meaning of the statement [latex]g(100)=1[/latex].

71. Let [latex]f(t)[/latex] be the number of ducks in a lake [latex]t[/latex] years after 1990. Explain the meaning of each statement:
[latex]f(5)=30[/latex]
[latex]f(10)=40[/latex]

72. Let [latex]h(t)[/latex] be the height above ground, in feet, of a rocket [latex]t[/latex] seconds after launching. Explain the meaning of each statement:
[latex]h(1)=200[/latex]
[latex]h(2)=350[/latex]

73. Show that the function [latex]f(x)=3(x−5)^2+7[/latex] is not one-to-one.

For the following exercises, find the domain of each function using interval notation.

74. [latex]f\left(x\right)=-2x\left(x - 1\right)\left(x - 2\right)[/latex]

75. [latex]f\left(x\right)=5 - 2{x}^{2}[/latex]

76. [latex]f\left(x\right)=3\sqrt{x - 2}[/latex]

77. [latex]f\left(x\right)=3-\sqrt{6 - 2x}[/latex]

78. [latex]f\left(x\right)=\sqrt{4 - 3x}[/latex]

79. [latex]f\left(x\right)=\sqrt{{x}^{2}+4}[/latex]

80. [latex]f\left(x\right)=\sqrt[3]{1 - 2x}[/latex]

81. [latex]f\left(x\right)=\sqrt[3]{x - 1}[/latex]

82. [latex]f\left(x\right)=\frac{9}{x - 6}[/latex]

83. [latex]f\left(x\right)=\frac{3x+1}{4x+2} [/latex]

84. [latex]f\left(x\right)=\frac{\sqrt{x+4}}{x - 4} [/latex]

85. [latex]f\left(x\right)=\frac{x - 3}{{x}^{2}+9x - 22} [/latex]

86. [latex]f\left(x\right)=\frac{1}{{x}^{2}-x - 6} [/latex]

87. [latex]f\left(x\right)=\frac{2{x}^{3}-250}{{x}^{2}-2x - 15} [/latex]

88. [latex]\frac{5}{\sqrt{x - 3}} [/latex]

89. [latex]\frac{2x+1}{\sqrt{5-x}} [/latex]

90. [latex]f\left(x\right)=\frac{\sqrt{x - 4}}{\sqrt{x - 6}} [/latex]

91. [latex]f\left(x\right)=\frac{\sqrt{x - 6}}{\sqrt{x - 4}} [/latex]

92. [latex]f\left(x\right)=\frac{x}{x} [/latex]

93. [latex]f\left(x\right)=\frac{{x}^{2}-9x}{{x}^{2}-81} [/latex]

For exercises 94-99, find the difference quotient for the given function.

94. [latex]f\left(x\right)=-7[/latex]

95. [latex]f\left(x\right)=4x-2[/latex]

96. [latex]f\left(x\right)=2{x}^{2}-3x[/latex]

97. [latex]f\left(x\right)=2{x}^{2}-5x+1[/latex]

98. [latex]f\left(x\right)=3{x}^{3}[/latex]

99. [latex]f\left(x\right)=\frac{1}{x+3}[/latex]