Learning Objectives
In this section, you will:
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.
1.2.1 – Determining Whether a Relation Represents a Function
A relation is a set of ordered pairs. The set consisting of the first components of each ordered pair is called the domain and the set consisting 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, or independent variable, and is often labeled with the lowercase letter [latex]\,x.\,[/latex] Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter [latex]\,y.[/latex]
A function [latex]\,f\,[/latex] is a relation that assigns a single element in the range to each element 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.
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.
- Identify the input values.
- Identify the output values.
- 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 in Figure 2 consists of items and their prices.
- 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.
- 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.
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 figure 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 |
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 figure below lists the five greatest baseball players of all time in order of rank[1].
| Player | Rank |
|---|---|
| Babe Ruth | 1 |
| Willie Mays | 2 |
| Ty Cobb | 3 |
| Walter Johnson | 4 |
| Hank Aaron | 5 |
- Is the rank a function of the player name?
- Is the player name a function of the rank?
Show answer
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 graphing calculators and 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.
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.
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\text{, }[/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. Assume that the domain does not include leap years.
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.
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
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?
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.
Try It
The weight of a pig, [latex]W[/latex], is a function of its age in days, [latex]d[/latex]. That is, [latex]W = f(d)[/latex]. What does [latex]15=f\left(21\right)[/latex] tell you about the pig?
Show answer
When the pig is 21 days old, it will weigh 15 pounds.
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\text{\hspace{0.17em}.}[/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.
- Identify the input and output values.
- 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 below represents a function (if any)?
| Input | Output |
|---|---|
| 2 | 1 |
| 5 | 3 |
| 8 | 6 |
| Input | Output |
|---|---|
| –3 | 5 |
| 0 | 1 |
| 4 | 5 |
| Input | Output |
|---|---|
| 1 | 0 |
| 5 | 2 |
| 5 | 4 |
Table 1 and Table 2 define functions. In both, each input value corresponds to exactly one output value. Table 3 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 1 can be represented by writing
Similarly, the statements
represent the function in Table 2.
Table 3 cannot be expressed in a similar way because it does not represent a function.
Try It
Does the table below represent a function?
| Input | Output |
|---|---|
| 1 | 10 |
| 2 | 100 |
| 3 | 1000 |
Show answer
yes
1.2.2 – Finding Input and Output Values of a Function
In section 1.1.1, we said that we evaluate an “algebraic expression by plugging in particular values for each of the variables and then simplifying the resulting expression using the order of operations.” In the same way, we can evaluate a function. That is, if we know the input then we can plug it into our function to determine the output. Evaluating a function 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 [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.
- Replace the input variable in the formula with the value provided.
- Calculate the result.
Example 6 – Evaluating Functions at Specific Values
Evaluate the function [latex]\,f\left(x\right)={x}^{2}+3x-4\,[/latex] at
- [latex]x = 2[/latex]
- [latex]x = u[/latex]
To evaluate the function, we replace the [latex]\,x\,[/latex] in the formula with each specified value.
- Because the input value is a number, 2, we can use simple algebra to simplify.
[latex]$$\begin{array}{ccc}\hfill f\left(2\right)& =& {2}^{2}+3\left(2\right)-4\\ & =& 4+6-4\hfill \\ & =& 6\hfill \end{array}$$[/latex]
- In this case, the input value is a letter so we cannot simplify the answer any further.
[latex]$$f\left(u\right)={u}^{2}+3u-4$$[/latex]
Example 7 – Evaluating Functions
Given the function [latex]\,h\left(p\right)={p}^{2}+2p,\,[/latex] evaluate [latex]\,h\left(4\right).\,[/latex]
To evaluate [latex]\,h\left(4\right),\,[/latex] we substitute the value 4 for the input variable [latex]\,p\,[/latex] in the given function.
Therefore, for an input of 4, we have an output of 24.
Try It
Given the function [latex]\,g\left(m\right)=\sqrt{m-4},\,[/latex] evaluate [latex]\,g\left(5\right).[/latex]
Show answer
[latex]\,g\left(5\right)=1\,[/latex]
Example 8 – Solving an Equation to Find a Specific Input to a Function
Consider the function [latex]\,h\left(p\right)={p}^{2}+2p[/latex]. What input to this function would give us the output 3? Can you set up an equation to find this input?
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.
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] We can also verify by graphing as in Figure 4. 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)=4m-7,\,[/latex] find an input [latex]m[/latex] that gives us the output [latex]\,g\left(m\right)=5.[/latex]
Show answer
We set up the equation [latex]4m - 7 = 5[/latex] and solve for [latex]m[/latex]. Adding [latex]7[/latex] to both sides, we get
[latex]$$4m = 12 $$[/latex]
Dividing both sides by [latex]4[/latex] gives us
[latex]$$m = 3 $$[/latex]
Note, we can check this answer by plugging [latex]3[/latex] back into our function to confirm that [latex]g\left(3\right) = 4\cdot 3 - 7 = 5[/latex].
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[2].
| Pet | Memory span in hours |
|---|---|
| Puppy | 0.008 |
| Adult dog | 0.083 |
| Cat | 16 |
| Goldfish | 2160 |
| Beta fish | 3600 |
At times, representing 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.
- Find the given input in the row (or column) of input values.
- Identify the corresponding output value paired with that input value.
- Find the given output values in the row (or column) of output values, noting every time that output value appears.
- Identify the input value(s) corresponding to the given output value.
Example 9 – Working with Functions Represented in Tables
Using the table below,
- Evaluate [latex]\,g\left(3\right).[/latex]
- Find an input [latex]n[/latex] such that [latex]\,g\left(n\right)=6.[/latex]
| [latex]n[/latex] | 1 | 2 | 3 | 4 | 5 |
| [latex]g\left(n\right)[/latex] | 8 | 6 | 7 | 6 | 8 |
- 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 shows two solutions: [latex]2[/latex] and [latex]4.[/latex]
| [latex]n[/latex] | 1 | 2 | 3 | 4 | 5 |
| [latex]g\left(n\right)[/latex] | 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 from Example 9, evaluate [latex]\,g\left(1\right).[/latex]
Show answer
[latex]g\left(1\right)=8[/latex]
Working with Graphs of Functions
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 10 – Reading Function Values from a Graph
Given the graph in Figure 5,
- To evaluate [latex]\,f\left(2\right),\,[/latex] locate the point on the curve where [latex]\,x=2,\,[/latex] then read the y-coordinate of that point. The point has coordinates [latex]\,\left(2,1\right),\,[/latex] so [latex]\,f\left(2\right)=1.\,[/latex] See (Figure 6).
- To solve [latex]\,f\left(x\right)=4,\,[/latex] we find the output value [latex]\,4\,[/latex] on the vertical axis. Moving horizontally along the line [latex]\,y=4,\,[/latex] we locate two points of the curve with output value [latex]\,4:[/latex] [latex]\left(-1,4\right)\,[/latex] and [latex]\,\left(3,4\right).\,[/latex] These points represent the two solutions to [latex]\,f\left(x\right)=4:[/latex] [latex]-1\,[/latex] or [latex]\,3.\,[/latex] This means [latex]\,f\left(-1\right)=4\,[/latex] and [latex]\,f\left(3\right)=4,\,[/latex] or when the input is [latex]\,-1\,[/latex] or [latex]\text{3,}\,[/latex] the output is [latex]\,\text{4}\text{.}\,[/latex] See (Figure 7).
Try It
Using Figure 7, solve [latex]\,f\left(x\right)=1.[/latex]
Show answer
[latex]x=0\,[/latex] or [latex]\,x=2\,[/latex]
Access the following online resources for additional instruction and practice with functions.
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. See Example 1 and Example 2.
- Function notation is a shorthand method for relating the input to the output in the form [latex]\,y=f\left(x\right).\,[/latex] See Example 3 and Example 4.
- In tabular form, a function can be represented by rows or columns that relate to input and output values. See Example 5.
- 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. See Example 6 and Example 7.
- To solve for a specific function value, we determine the input values that yield the specific output value. See Example 8.
- Input and output values of a function can be identified from a table. See Example 9.
- Relating input values to output values on a graph is another way to evaluate a function. See Example 10.
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