Defining and Writing Functions

Learning Objectives

In this section, you will:

• Determine whether a relation represents a function.
• Find the value of a function.
• Determine whether a function is one-to-one.
• Use the vertical line test to identify functions.
• Graph the functions listed in the library of functions.

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.

$\{(1,2), \quad(2,4), \quad(3,6), \quad(4,8), \quad(5,10)\}$

The domain is $\{1,2,3,4,5\}$. The range is $\{2,4,6,8,10\}$.

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter $x$. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter $y$.

A function $f$ 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, $\{1,2,3,4,5\}$, is paired with exactly one element in the range, $\{2,4,6,8,10\}$.

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

$\{(\operatorname{odd}, 1), \quad(\text { even, 2), }(\text { odd }, 3), \quad(\text { even }, 4), \quad(\text { odd }, 5)\}$

Notice that each element in the domain, $\{$ even, odd $\}$ is not paired with exactly one element in the range, $\{1,2,3,4,5\}$. For example, the term "odd" corresponds to three values from the range, $\{1,3,5\}$ and the term "even" corresponds to two values from the range, $\{2,4\}$. 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 $q$ and $r$ both give output $n$. (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 $q$ is associated with two different outputs.

FUNCTION

A function is a relation in which each possible input value leads to exactly one output value. We say "the output is a function of the input".

The input values make up the domain, and the output values make up the range.

HOW TO

Given a relationship between two quantities, determine whether the relationship is a function.

1. Identify the input values.

2. Identify the output values.

3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

EXAMPLE 1

Determining If Menu Price Lists Are Functions

The coffee shop menu, shown below, consists of items and their prices.

a) Is price a function of the item?

b) Is the item a function of the price?

Solution

a) Let's begin by considering the input as the items on the menu. The output values are then the prices.

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

b) 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 the image below.

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? Table 1 shows a possible rule for assigning grade points.

Table 1

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 #1

Table 2 lists the five greatest baseball players of all time in order of rank.

Table 2

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

b) Is the player name a function of the 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 $h$ for height and $a$ for age. The letters $f, g$, and $h$ are often used to represent functions just as we use $x, y$, and $z$ to represent numbers and $A, \quad B$, and $C$ to represent sets.

$\begin{array}{lll} h \text { is } f \text { of } a & \text { We name the function } f ; \text { height is a function of age. } \\ h=f(a) & \text { We use parentheses to indicate the function input. } \\ f(a) & \text { We name the function } f ; \text { the expression is read as "} f \, \text {of } \, a". \end{array}$

Remember, we can use any letter to name the function; the notation $h(a)$ shows us that $h$ depends on $a$. The value $a$ must be put into the function $h$ 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 $f(a+b)$ 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.

FUNCTION NOTATION

The notation $y=f(x)$ defines a function named $f$. This is read as " $y$ is a function of $x$". The letter $x$ represents the input value, or independent variable. The letter $y$, or $f(x)$, 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. Assume that the domain does not include leap years.

Solution

The number of days in a month is a function of the name of the month, so if we name the function $f$, we write days $=f($ month $)$ or $d=f(m)$. The name of the month is the input to a "rule" that associates a specific number (the output) with each input.

Figure 2

For example, $f($ March $)=31$, because March has 31 days. The notation $d=f(m)$ reminds us that the number of days, $d$ (the output), is dependent on the name of the month, $m$ (the input).

Analysis

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 $N=f(y)$ gives the number of police officers, $N$, in a town in year $y$. What does $f(2005)=300$ represent?

Solution

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

TRY IT #2

Use function notation to express the weight of a pig in pounds as a function of its age in days $d$.

Q&A

Instead of a notation such as $y=f(x)$, could we use the same symbol for the output as for the function, such as $y=y(x)$, 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 $f$, which is a rule or procedure, and the output $y$ we get by applying $f$ to a particular input $x$. This is why we usually use notation such as $y=f(x), P=W(d)$, 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.

Table 3 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 $f$ where $D=f(m)$ identifies months by an integer rather than by name.

Table 3

Table 4 defines a function $Q=g(n)$. Remember, this notation tells us that $g$ is the name of the function that takes the input $n$ and gives the output $Q$.

Table 4

Table 5 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.

Table 5

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, Table 6, Table 7, or Table 8, represents a function (if any)?

Table 6

Table 7

Table 8

Solution

Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 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 Table 6 can be represented by writing

$f(2)=1, f(5)=3, \text { and } f(8)=6$

Similarly, the statements

$g(-3)=5, \quad g(0)=1, \text { and } g(4)=5$

represent the function in Table 7.

Table 8 cannot be expressed in a similar way because it does not represent a function.

TRY IT #3

Does Table 9 represent a function?

Table 9

Source: Rice University, https://openstax.org/books/college-algebra/pages/3-1-functions-and-function-notation#0
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

Evaluation of 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 $f(x)=5-3 x^{2}$ 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. Substitute the input variable in the formula with the value provided.

2. Calculate the result.

EXAMPLE 6

Evaluating Functions at Specific Values

Evaluate $f(x)=x^{2}+3 x-4$ at:

(a) 2

(b) $a$

(c) $a+h$

(d) Now evaluate $\dfrac{f(a+h)-f(a)}{h}$

Solution

Replace the $x$ in the function with each specified value.

(a) Because the input value is a number, 2, we can use simple algebra to simplify.

$\begin{aligned} f(2)=2^{2}+3(2)-& 4 \\ &=4+6-4 \\ &=6 \end{aligned}$

(b) In this case, the input value is a letter so we cannot simplify the answer any further.

$f(a)=a^{2}+3 a-4$

With an input value of $a+h$, we must use the distributive property.

$\begin{aligned} f(a+h)=(a+h)^{2}+3(a+h)-4 & \\ &=a^{2}+2 a h+h^{2}+3 a+3 h-4 \end{aligned}$

(c) 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

$f(a+h)=a^{2}+2 a h+h^{2}+3 a+3 h-4$

and we know that

$f(a)=a^{2}+3 a-4$

Now we combine the results and simplify.

$\begin{aligned}\frac{f(a+h)-f(a)}{h} &=\frac{\left(a^{2}+2 a h+h^{2}+3 a+3 h-4\right)-\left(a^{2}+3 a-4\right)}{h}\\&=\frac{2 a h+h^{2}+3 h}{h}\\&=\frac{h(2 a+h+3)}{h} \qquad \qquad \text { Factor out } h \text {. }\\&=2 a+h+3 \qquad \qquad \qquad \text { Simplify. }\end{aligned}$

EXAMPLE 7

Evaluating Functions

Given the function $h(p)=p^{2}+2 p$, evaluate $h(4)$.

Solution

To evaluate $h(4)$, we substitute the value 4 for the input variable $p$ in the given function.

$\begin{aligned} h(p) &=p^{2}+2 p \\ h(4) &=(4)^{2}+2(4) \\ &=16+8 \\ &=24 \end{aligned}$

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

TRY IT #4

Given the function $g(m)=\sqrt{m-4}$, evaluate $g(5)$.

EXAMPLE 8

Solving Functions

Given the function $h(p)=p^{2}+2 p$, solve for $h(p)=3$.

Solution

$\begin{aligned} h(p)=3 & \\ p^{2}+2 p=3 & \text { Substitute the original function } h(p)=p^{2}+2 p \\ p^{2}+2 p-3=0 & \text { Subtract } 3 \text { from each side. } \\(p+3)(p-1)=0 & \text { Factor. } \end{aligned}$

If $(p+3)(p-1)=0$, either $(p+3)=0$ or $(p-1)=0$ (or both of them equal 0). We will set each factor equal to 0 and solve for $p$ in each case.

$\begin{aligned} &(p+3)=0, \quad p=-3 \\ &(p-1)=0, \quad p=1 \end{aligned}$

This gives us two solutions. The output $h(p)=3$ when the input is either $p=1$ or $p=-3$. We can also verify by graphing as in Figure 3. The graph verifies that $h(1)=h(-3)=3$ and $h(4)=24$.

Figure 3

TRY IT #5

Given the function $g(m)=\sqrt{m-4}$, solve $g(m)=2$.

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 $2 n+6 p=12$ expresses a functional relationship between $n$ and $p$. We can rewrite it to decide if $p$ is a function of $n$.

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 $2 n+6 p=12$ as a function $p=f(n)$, if possible.

Solution

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

$2 n+6 p=12$

$\begin{array}{ll} 6 p=12-2 n & \text { Subtract } 2 n \text { from both sides. } \\ p=\dfrac{12-2 n}{6} & \text { Divide both sides by } 6 \text { and simplify. } \\ p=\dfrac{12}{6}-\dfrac{2 n}{6} & \\ p=2-\dfrac{1}{3} n & \end{array}$

Therefore, $p$ as a function of $n$ is written as

$p=f(n)=2-\dfrac{1}{3} n$

Analysis

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 $x^{2}+y^{2}=1$ represent a function with $x$ as input and $y$ as output? If so, express the relationship as a function $y=f(x)$.

Solution

First we subtract $x^{2}$ from both sides.

$y^{2}=1-x^{2}$

We now try to solve for $y$ in this equation.

$\begin{aligned} y &=\pm \sqrt{1-x^{2}} \\ &=+\sqrt{1-x^{2}} \quad \text { and }-\sqrt{1-x^{2}} \end{aligned}$

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $y=f(x)$.

TRY IT #6

If $x-8 y^{3}=0$, express $y$ as a function of $x$.

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 $x=y+2^{y}$, if we want to express $y$ as a function of $x$, there is no simple algebraic formula involving only $x$ that equals $y$. However, each $x$ does determine a unique value for $y$, and there are mathematical procedures by which $y$ can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for $y$ as a function of $x$, 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 Table 10.

Table 10

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function $P$. 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 $P$ at the input value of "goldfish". We would write $P($ goldfish $)=2160$. 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 $P$ 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 Table 11,

(a) Evaluate $g(3)$.

(b) Solve $g(n)=6$.

Table 11

Solution

(a) Evaluating $g(3)$ means determining the output value of the function $g$ for the input value of $n=3$. The table output value corresponding to $n=3$ is 7, so $g(3)=7$.

(b) Solving $g(n)=6$ means identifying the input values, $n$, that produce an output value of 6. Table 11 shows two solutions: $2$ and $4$.

When we input 2 into the function $g$, our output is 6. When we input 4 into the function $g$, our output is also 6.

TRY IT #7

Using [link], evaluate $g(1)$.

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

EXAMPLE 12

Reading Function Values from a Graph

Given the graph in Figure 4,

(a) Evaluate $f(2)$.

(b) Solve $f(x)=4$.

Figure 4

Solution

(a) To evaluate $f(2)$, locate the point on the curve where $x=2$, then read the $y$-coordinate of that point. The point has coordinates $(2,1)$, so $f(2)=1$. See Figure 5.

Figure 5

(b) To solve $f(x)=4$, we find the output value $4$ on the vertical axis. Moving horizontally along the line $y=4$, we locate two points of the curve with output value $4:(-1,4)$ and $(3,4)$. These points represent the two solutions to $f(x)=4:-1$ or $3$. This means $f(-1)=4$ and $f(3)=4$, or when the input is $-1$ or $3$, the output is $4$. See Figure 6.

Figure 6

TRY IT #8

Using Figure 4, solve $f(x)=1$.