Topics
4. Functions and Relations
topic badge
United States of AmericaVA
Algebra, Functions, and Data Analysis

4.03 Evaluating functions

Lesson
Practice

Evaluating functions

Recall that a function maps each input of a relation to exactly one output. Functions are typically represented in function notation, which makes the relationship between inputs and outputs 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.

Function notation is also useful for representing real-world relationships, where x and f\left(x\right) represent specific quantities.

We have seen the equation for a linear function y=mx+b. By naming a linear function f, we can also write the function using function notation: f\left(x\right)=mx+b.

This is a way of saying that mx+b is a function of x. Using f(x) clearly shows we're dealing with a function, unlike just using y, which could sometimes represent equations that aren't functions.

The notation f\left(x\right) is another name for y. If f is a function and x is an allowed input (part of its domain), then f(x) is the specific output obtained from that input x.

Letters other than f can be used to name a function, such as g or h.

\displaystyle f\left(x\right)=y
\bm{f}
is the name of function
\bm{x}
is the input
\bm{y}
is the output

The graph of a function f consists of all points \left(x, y\right) where y = f\left(x\right). Therefore, if a point \left(a, b\right) lies on the graph of f, it means that f\left(a\right) = b.

To evaluate a function means to find the output value for a specific input value.

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

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

\begin{aligned} f\left(1\right) &= -7 \cdot \left(1\right)+9 \\ &= -7+9 \\ &= 2 \end{aligned}

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

Examples

Example 1

For the functionf\left(x\right) = \dfrac{x}{3}-5where x is the independent variable.

a

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

Worked Solution
Create a strategy

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) & = \dfrac{-3}{3} - 5 \\ & = -6 \end{aligned}

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

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

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

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

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

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

Evaluate the function for f\left(2\right).

Worked Solution
Apply the idea

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

Example 2

-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f(x)
a

Evaluate the function for f\left(-1\right).

Worked Solution
Create a strategy

To find f(-1) from the graph, locate x=-1 on the x-axis. Then, move vertically to the point on the function's curve. Finally, move horizontally from that point to the y-axis to read the output value.

Apply the idea

Locate x=-1 on the x-axis and move vertically down to the curve. Then, move horizontally to the y-axis.

-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f(x)

When x=-1, the point on the graph is (-1, -2), so f(-1) = -2.

b

Determine the value of x when f\left(x\right)=4.

Worked Solution
Create a strategy

To find the value of x when f(x)=4, locate y=4 on the y-axis. Then, move horizontally to the point on the function's curve. Finally, move vertically from that point down to the x-axis to read the input value.

Apply the idea

Locate y=4 on the y-axis and move horizontally left to the curve. Then, move vertically down to the x-axis.

-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f(x)

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

Example 3

For the function f\left(x\right)=3x-5:

a

Find the range when the domain is \{-3, 0, 11\}.

Worked Solution
Create a strategy

We need to find the range (the values of f\left(x\right)) when the domain (the x-values) is \{-3,0,11\}. Substitute x-values and solve for f\left(x\right).

Apply the idea

Substitute x=-3 for x in the function.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(-3\right)\displaystyle =\displaystyle 3 \cdot \left(-3\right)-5Substitute x=-3
\displaystyle {}\displaystyle =\displaystyle -9-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle -14Evaluate the subtraction

Substitute x=0 for x in the function.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(0\right)\displaystyle =\displaystyle 3 \cdot \left(0\right)-5Substitute x=0
\displaystyle {}\displaystyle =\displaystyle 0-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle -5Evaluate the subtraction

Substitute x=11 for x in the function.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(11\right)\displaystyle =\displaystyle 3 \cdot \left(11\right)-5Substitute x=11
\displaystyle {}\displaystyle =\displaystyle 33-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 28Evaluate the subtraction

When the domain is \{-3,0,11\}, this function has a range of \{-14,-5,28\}.

Reflect and check

This could be written in function notation as \{f\left(-3\right),\,f\left(0\right),\, f\left(11\right)\}=\{-14,\,-5,\,28\}.

b

Find the domain when the range is \{-2,4,7\}.

Worked Solution
Create a strategy

We need to find the domain (the x-values) when the range (the values of f\left(x\right)) is \{-2,4,7\}. Substitute the f\left(x\right) values and solve for x.

Apply the idea

Replace f\left(x\right) with -2 in the equation and solve for x.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle -2\displaystyle =\displaystyle 3x-5Substitute f\left(x\right)=-2
\displaystyle 3\displaystyle =\displaystyle 3xAdd 5 to both sides
\displaystyle 1\displaystyle =\displaystyle xDivide both sides by 3

Replace f\left(x\right) with 4 in the equation and solve for x.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle 4\displaystyle =\displaystyle 3x-5Substitute f\left(x\right)=4
\displaystyle 9\displaystyle =\displaystyle 3xAdd 5 to both sides
\displaystyle 3\displaystyle =\displaystyle xDivide both sides by 3

Replace f\left(x\right) with 7 in the equation and solve for x.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle 7\displaystyle =\displaystyle 3x-5Substitute f\left(x\right)=7
\displaystyle 12\displaystyle =\displaystyle 3xAdd 5 to both sides
\displaystyle 4\displaystyle =\displaystyle xDivide both sides by 3

When this function has a range of \{-2,\,4,\,7\}, the domain is \{1,\,3,\,4\}.

Reflect and check

This means f(1) = -2, f(3) = 4, and f(4) = 7.

c

Evaluate f(7) - f(2).

Worked Solution
Create a strategy

Find the values of f\left(7\right) and f\left(2\right), then subtract the value of f\left(2\right) from the value of f\left(7\right).

Apply the idea

First, let's evaluate f\left(7\right).

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(7\right)\displaystyle =\displaystyle 3 \cdot \left(7\right)-5Substitute x=7
\displaystyle {}\displaystyle =\displaystyle 21-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 16Evaluate the subtraction

Now, evaluate f(2).

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(2\right)\displaystyle =\displaystyle 3 \cdot \left(2\right)-5Substitute x=2
\displaystyle {}\displaystyle =\displaystyle 6-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 1Evaluate the subtraction

Now, we can find f\left(7\right)-f\left(2\right) by substituting their values.\begin{aligned} f\left(7\right)-f\left(2\right) &= 16-1 \\ &= 15 \end{aligned}

Example 4

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
0
2
4
6
8
10
12
14
16
18
20
\text{Time (days)}
2
4
6
8
10
12
14
16
18
20
\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
0
2
4
6
8
10
12
14
16
18
20
\text{Time (days)}
2
4
6
8
10
12
14
16
18
20
\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
0
2
4
6
8
10
12
14
16
18
20
\text{Time (days)}
2
4
6
8
10
12
14
16
18
20
\text{Height (inches)}
Idea summary

An equation where the output variable is isolated, like y=mx+b, can be written as a function in the form f\left(x\right)=mx+b. We evaluate a function, written in function notation as f\left(c\right), by replacing all values of x with c and evaluating the expression. In real-world problems, f\left(x\right) represents a specific quantity that depends on the input quantity x.

Remember that any point \left(x, y\right) on the graph of function f represents an input x and its corresponding output y = f\left(x\right).

\displaystyle f\left(x\right)=y
\bm{f}
is the name of function
\bm{x}
is the input
\bm{y}
is the output

Outcomes

AFDA.AF.2e

Describe and recognize the connection between points on the graph and the value of a function.

What is Mathspace

About Mathspace