Function Notation: How to Evaluate f(x)

Function notation is a way to name a function and show its input. In $f(x) = 2x + 3$ , $f$ is the name of the function, $x$ is the input, and $f(x)$ is the output. To evaluate $f(4)$ , replace $x$ with 4: $f(4) = 2(4) + 3 = 11$ .

Before this lesson: You should know what a function is from relations and functions, and how to evaluate algebraic expressions.

Extra practice: The Algebra 1 Unit 3 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Does f(x) Mean?

You read $f(x)$ as “f of x.” It does not mean $f$ times $x$ . It means “the output of the function $f$ when the input is $x$ .”

Think of a function as a machine. You drop in an input. The machine follows its rule. Out comes one output. The notation $f(4)$ asks, “What comes out when 4 goes in?”

A function machine. The input 4 goes into a box labeled f with the rule multiply by 2, then add 3. The output 11 comes out. Below it reads f(4) = 11. — Put 4 into the machine f(x) = 2x + 3 and 11 comes out, so f(4) = 11.

Function notation is just a label for $y$ . The equation $y = 2x + 3$ and the function $f(x) = 2x + 3$ make the same line. The fact $f(4) = 11$ means the point $(4, 11)$ is on the graph.

Why use it? The name tells you which function you mean, and the input is right there. If a problem has two functions, you can call them $f$ and $g$ and never mix them up.

How to Evaluate a Function

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

Write the rule. Replace every $x$ with the input. Put the input in parentheses.
Simplify with the order of operations.
Write the answer as $f(\text{input}) = \text{output}$ .

Some problems go the other way. They give the output and ask for the input. For “Find $x$ when $f(x) = 19$ ,” set the rule equal to 19 and solve the equation.

The question	What it asks	What you do
Find $f(3)$	The output when the input is 3	Replace $x$ with 3 and simplify
Solve $f(x) = 3$	The input that gives an output of 3	Set the rule equal to 3 and solve

Function Notation Examples

Example 1: Evaluate a linear function

Let $f(x) = 3x - 5$ . Find $f(4)$ .

f(4) = 3(4) - 5

Step 1. Replace $x$ with 4.

f(4) = 12 - 5

Step 2. Multiply first, following the order of operations.

f(4) = 7

Step 3. Subtract.

Answer: $f(4) = 7$ .

Example 2: A negative input and a square

Let $g(x) = x^2 - 2x + 1$ . Find $g(-3)$ .

g(-3) = (-3)^2 - 2(-3) + 1

Step 1. Replace both $x$ ‘s with $(-3)$ . The parentheses keep the sign with the number.

g(-3) = 9 - 2(-3) + 1

Step 2. Square first: $(-3)^2 = 9$ . A negative times a negative is positive. (See multiplying and dividing integers.)

g(-3) = 9 + 6 + 1

Step 3. Multiply: $-2(-3) = 6$ .

g(-3) = 16

Step 4. Add from left to right.

Answer: $g(-3) = 16$ .

Example 3: Find the input from the output

Let $h(x) = 4x + 7$ . Find $x$ when $h(x) = 19$ .

4x + 7 = 19

Step 1. Replace $h(x)$ with its rule. Now you have an equation to solve, like the ones in two-step equations.

4x = 12

Step 2. Subtract 7 from both sides.

x = 3

Step 3. Divide both sides by 4.

Answer: $x = 3$ .

Check: $h(3) = 4(3) + 7 = 12 + 7 = 19$ . It works.

Example 4: Read a function from a graph

The graph shows $f(x) = 2x - 1$ . Find $f(2)$ . Then find $x$ when $f(x) = -3$ .

The line f(x) = 2x − 1 on a grid from −5 to 5. A dashed path goes up from x = 2 on the x-axis to the point (2, 3) on the line, then left to 3 on the y-axis. The point (−1, −3) is also marked. — Go up from x = 2 to the line, then across to the y-axis: f(2) = 3.

Step 1. To find $f(2)$ , start at 2 on the $x$ -axis.

f(2) = 3

Step 2. Go straight up to the line. You land at the point (2, 3), so the height is 3.

f(2) = 2(2) - 1 = 4 - 1 = 3

Step 3. Check with the rule.

Step 4. To solve $f(x) = -3$ , start at $-3$ on the $y$ -axis.

Step 5. Go straight across to the line. It meets the line at the point $(-1, -3)$ , so $x = -1$ .

f(-1) = 2(-1) - 1 = -2 - 1 = -3

Step 6. Check with the rule.

Answer: $f(2) = 3$ , and $f(x) = -3$ when $x = -1$ .

Example 5: Word problem

A plumber charges a $60 trip fee plus $45 per hour. The cost for $h$ hours is $C(h) = 45h + 60$ dollars. Find $C(3)$ and explain what it means. Then find how many hours cost $285.

C(3) = 45(3) + 60

Step 1. Replace $h$ with 3.

C(3) = 135 + 60

Step 2. Multiply.

C(3) = 195

Step 3. Add.

Step 4. Explain it. $C(3) = 195$ means a 3-hour job costs $195.

45h + 60 = 285

Step 5. To find the hours for $285, set the rule equal to 285.

45h = 225

Step 6. Subtract 60 from both sides.

h = 5

Step 7. Divide both sides by 45.

Answer: A 3-hour job costs $195. A bill of $285 is for 5 hours of work.

Check: $C(5) = 45(5) + 60 = 225 + 60 = 285$ .

Common Mistakes

Mistake	Example of the error	Fix
Reading $f(x)$ as multiplication	Writing $f(2) = 2f$	$f(2)$ means the output when the input is 2.
Leaving out parentheses with a negative input	For $x^2$ at $x = -3$ , writing $-3^2 = -9$	Write $(-3)^2 = 9$ .
Mixing up $f(3)$ and $f(x) = 3$	Plugging in 3 when asked to solve $f(x) = 3$	$f(x) = 3$ gives the output. Set the rule equal to 3 and solve.
Replacing only one $x$	For $g(x) = x^2 - 2x$ , finding $g(4) = 16 - 2x$	Replace every $x$ with the input.

Function Notation in Tables and Other Letters

A function can be given as a table. To find $f(1)$ , find 1 in the $x$ row and read the output below it.

$x$	$-1$	0	1	2
$f(x)$	4	1	$-2$	$-5$

Here $f(1) = -2$ . To solve $f(x) = 4$ , find 4 in the bottom row. Its input is $-1$ , so $x = -1$ .

Functions don’t have to use $f$ and $x$ . Word problems often use letters that match the meaning, like $C(h)$ for cost after $h$ hours or $d(t)$ for distance after $t$ seconds. The steps are the same.

You will use function notation for lines in slope-intercept form: $f(x) = mx + b$ is the same line as $y = mx + b$ .

Practice Problems

Use $f(x) = 5x + 2$ , $g(x) = x^2 - 4$ and $h(x) = -2x + 9$ for problems 1–7.

Find $f(3)$ .
Find $f(-2)$ .
Find $g(5)$ .
Find $g(-1)$ .
Find $h(0)$ .
Find $x$ when $f(x) = 27$ .
Find $x$ when $h(x) = 15$ .
Let $p(x) = \frac{x}{2} + 3$ . Find $p(-6)$ .
Use the table in the section above. Find $f(1)$ , and find $x$ when $f(x) = 4$ .
A phone plan costs $C(m) = 0.10m + 15$ dollars for $m$ minutes. Find $C(200)$ . How many minutes cost $40?

Show answers

f(3) = 5(3) + 2

Substitute 3.

f(3) = 15 + 2

Multiply.

f(3) = 17

Add.

f(-2) = 5(-2) + 2

Substitute −2.

f(-2) = -10 + 2

Multiply.

f(-2) = -8

Add.

g(5) = 5^2 - 4

Substitute 5.

g(5) = 25 - 4

Square.

g(5) = 21

Subtract.

g(-1) = (-1)^2 - 4

Substitute −1 in parentheses.

g(-1) = 1 - 4

Square.

g(-1) = -3

Subtract.

h(0) = -2(0) + 9

Substitute 0.

h(0) = 0 + 9

Multiply.

h(0) = 9

Add.

5x + 2 = 27

Set the rule equal to 27.

5x = 25

Subtract 2 from both sides.

x = 5

Divide both sides by 5.

-2x + 9 = 15

Set the rule equal to 15.

-2x = 6

Subtract 9 from both sides.

x = -3

Divide both sides by −2.

p(-6) = \frac{-6}{2} + 3

Substitute −6.

p(-6) = -3 + 3

Divide.

p(-6) = 0

Add.

f(1) = -2

Find 1 in the x row and read the output below it.

x = -1

Find 4 in the bottom row and read the input above it.

10.

C(200) = 0.10(200) + 15

Substitute 200.

C(200) = 20 + 15

Multiply.

C(200) = 35

Add.

So 200 minutes cost $35. For $40, set the rule equal to 40: $0.10m + 15 = 40$

0.10m = 25

Subtract 15 from both sides.

m = 250

Divide both sides by 0.10.

So $40 buys 250 minutes.

Want more practice? The Algebra 1 Unit 3 Practice Pack has 39 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Function Notation FAQ

What does f(x) mean?

$f(x)$ means the output of the function $f$ when the input is $x$ . You read it as “f of x.”

How do you evaluate a function?

Replace every $x$ in the rule with the input, using parentheses. Then simplify with the order of operations.

Is f(x) the same as y?

Yes, for graphing. The graph of $f(x) = 2x + 3$ is the same as the graph of $y = 2x + 3$ . Function notation also shows the input.

Does f(x) mean f times x?

No. The parentheses here hold the input. They do not mean multiplication.

Summary

$f(x)$ is read “f of x.” It is the output when the input is $x$ .
To find $f(a)$ , replace every $x$ with $a$ in parentheses, then simplify.
To solve $f(x) = c$ , set the rule equal to $c$ and solve for $x$ .
$f(a) = b$ means the point $(a, b)$ is on the graph.
Functions can use any letters, like C(h) for cost after h hours.

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