Topics
2. Linear functions
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United States of America
Grades 9-12

2.01 Evaluating functions

Lesson
Practice

Introduction

We first began exploring functions in 8th grade. Now, we will expand our understanding of how functions can be used to model real-world situations. We will formalize how we describe functions symbolically using function notation and practice interpreting or explaining function notation with respect to the relationships and quantities it describes.

Identifying and interpreting functions

Recall that a function maps each input of a relation to exactly one output. If an input matches to more than one output, the relation is not considered a function. Functions are typically represented in function notation, so the relationship between inputs and outputs are clear.

Input

The independent variable of a function; usually the x-value

Output

The dependent variable of a function; usually the y-value

Function notation

A notation that describes a function. For a function f when x is the input, the symbol f\left(x\right) denotes the corresponding output.

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We can use the vertical line test to help determine whether a relation is a function. When looking at a graph, if you can draw a vertical line anywhere so that it crosses the graph of the relation in more than one place, then it is not a function.

Here is an example of a relation that is not a function. See how when the blue vertical line is drawn in, it crossed the graph in two places?

Weekly Paycheck
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\text{Hours worked}, x
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\text{Total amount (in dollars)}, y

The graph depicts a relation where the independent variable is the hours worked and the dependent variable is the total amount in dollars. We can see that each value of y corresponds to a unique x-value, so the relation is a function and we can express y as f\left(x\right).

When x=4, then f(4)=60. This means a person who worked 4 hours earned \$ 60.

xy
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411
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722

The table shows the relationship between the number of laps a student runs, the independent variable x, and the total number of minutes, the dependent variable y. Since each output in the table corresponds to a unique input, the relation represents a function and we can express y as f\left(x\right).

f(5)=13 indicates that a student ran 5 laps in 13 minutes.

Examples

Example 1

Determine whether the following relations represent functions.

a
x-2-101234
y116323611
Worked Solution
Create a strategy

The relation is a function if for every x-value, there is exactly one y-value.

Apply the idea

We can see that each y-value corresponds to a unique x-value, so the relation is a function.

Reflect and check

We can see that the y-values, 11, 6, and 3 appear more than once in the table of values, but the relation is still a function. A relation is not a function only when multiple y-values point to the same x-value.

b
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x
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y
Worked Solution
Create a strategy

We can use the vertical line test to determine whether the relation is a function.

Apply the idea

Plotting a vertical line at x=1, we get the following:

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y

Since the vertical line passes through the relation twice at x=1, we determine the relation is not a function.

Reflect and check

We could have drawn a vertical line at any value of x that was greater than 0 and seen that the relation would not have passed the vertical line test.

c
Taxi fare
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\text{Miles driven}, x
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\text{Cost of ride (in dollars)}, y
Worked Solution
Create a strategy

We can use the vertical line test to determine whether the relation is a function.

Apply the idea

Plotting a vertical line at x=5, we get the following:

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\text{Miles driven}, x
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\text{Cost of ride (in dollars)}, y

Since the vertical line passes through the relation once at x=5, and every other value of x, we determine the relation is a function.

Example 2

Let f\left( x \right) represent the height of a growing plant, f, in inches, where x represents the time since it was planted in days.

Plant Growth
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\text{Time (days)}
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\text{Height (inches)}
a

Interpret the meaning of f\left(10\right) = 8.

Worked Solution
Create a strategy

We can use the units of the given information and the graph to help with the interpretation.

Apply the idea

We're given that f\left( x \right) represents the height of a growing plant in inches, so to interpret f\left( 10 \right), we need to determine what an input of x=10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted.

We also know that all of this is equal to 8. This is the output, or what our function f\left( x \right) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall.

Based on the graph, when x=10, y=8 so f(10)=8 is represented by the ordered pair (10,8) on the graph.

The plant has a height of 8 inches 10 days after being planted.

b

Interpret the meaning of f\left(6\right).

Worked Solution
Apply the idea

We know that x represents the time in days since the plant was planted and x=6. So this means that 6 days have passed since the plant was planted.

Since f\left( x \right) represents the height of a growing plant in inches, f\left( 6 \right) represents the height of the plant 6 days after being planted.

Reflect and check

Using the graph, we can find the actual height of the plant after 6 days.

Plant Growth
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\text{Time (days)}
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\text{Height (inches)}
c

Interpret the meaning of f\left(x\right)=12.

Worked Solution
Apply the idea

We know that f\left( x \right) represents the height of a growing plant in inches, so if f\left( x \right)=12, then the height of the plant is 12 inches x days after being planted.

Reflect and check

By using the graph, we can find the number of days when the height of the plant is 12 inches.

Plant Growth
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\text{Time (days)}
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\text{Height (inches)}
Idea summary

The notation y=f(x) defines a function named f. The variable x represents the input value of the function and f(x) represents the output. We can interpret function notation by matching the inputs and outputs of a function to the independent and dependent quantities of the context the function represents.

Evaluating function notation

To evaluate a function at a point is to calculate the output value at a particular input value:

If f(x)=-7x+9, then determine the value of f(1).

This is the same as stating to evaluate the function y=-7x+9 when x=1.

f(1)=-7(1)+9

f(1)=-7+9=2

Therefore, f(1)=2 for the function f(x)=-7x+9.

Examples

Example 3

Consider the equation x - 3y = 15

where x is the independent variable.

a

Rewrite the equation using function notation.

Worked Solution
Create a strategy

Since x is the independent variable, we want to rearrange the equation to isolate y, and then replace y with function notation. We can choose a symbol to represent the function, such as f.

Apply the idea

Rearranging the equation:

\displaystyle x-3y\displaystyle =\displaystyle 15Original equation
\displaystyle -3y\displaystyle =\displaystyle -x + 15Subtract x from both sides
\displaystyle y\displaystyle =\displaystyle \frac{x}{3} - 5Divide both sides by -3

We can now rewrite the equation using function notation as: f\left(x\right) = \frac{x}{3} - 5

b

Construct a table of values for the function at x=-3, \,0, \,9, \,12, \,27.

Worked Solution
Create a strategy

In order to construct a table of values, we will need to evaluate the function at the given values of x.

Apply the idea

Substituting x = -3, we have \begin{aligned} f\left(-3\right) & = \frac{-3}{3} - 5 \\ & = -6 \end{aligned}

Substituting x = 0, we have \begin{aligned} f\left(0\right) & = \frac{0}{3} - 5 \\ & = -5 \end{aligned}

Substituting x = 9, we have \begin{aligned} f\left(9\right) & = \frac{9}{3} - 5 \\ & = -2 \end{aligned}

Substituting x = 12, we have \begin{aligned} f\left(12\right) & = \frac{12}{3} - 5 \\ & = -1 \end{aligned}

Substituting x = 27, we have \begin{aligned} f\left(27\right) & = \frac{27}{3} - 5 \\ & = 4 \end{aligned}

A completed table for the function at the given values of x is

x-3091227
f\left( x \right)-6-5-2-14
c

Evaluate the function for f(2).

Worked Solution
Apply the idea

Substituting x =2, we have \begin{aligned} f\left(2\right) & = \frac{2}{3} - 5 \\ & = -\dfrac{13}{3} \end{aligned}

Example 4

Consider a table and graph that represent the same function:

x-2-1012
f\left( x \right)-7.5-6.75-6-5.25-4.5
-7
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x
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f(x)
a

Evaluate the function for f(-1).

Worked Solution
Create a strategy

We can either find the value of the function from the table or graph when x=-1.

Apply the idea

Since the value of the function is clearly stated in the table, we know that f(-1)=-6.75.

b

Determine the value of x when f(x)=-3.

Worked Solution
Create a strategy

We can either find the value of the function from the table or graph when f(x)=-3.

Apply the idea

When f(x)=-3, we can see on the graph that x=4.

Reflect and check

We can also continue to complete the table with the pattern in order to determine x when f(x)=-3.

x-2-101234
f\left( x \right)-7.5-6.75-6-5.25-4.5-3.75-3
Idea summary

An equation where the output variable is isolated like y=mx+b can be written as a function in the form, f(x)=mx+b. We evaluate a function, written in function notation as f(c), by replacing all values of x with c and evaluating the expression.

Outcomes

F.IF.A.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F.IF.A.2

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

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