Function Notation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Function Notation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Function notation $f(x)$ is a shorthand that names a function ( $f$ ) and specifies its input ( $x$ ). Writing $f(3) = 10$ means that when the input is 3, the function produces the output 10. This notation is not multiplication.

The notation $f(x)$ is not " $f$ times $x$ " — it means "the output of function $f$ when the input is $x$ ." The parentheses contain the input, not a multiplication.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $f(x)$ names a machine $f$ and the input $x$ you feed it, returning the output.

Common stuck point: The procedure for function notation is the easy part; the trap is reading $f(x)$ as $f$ times $x$ . Asking "Are the parentheses holding an input to a named function (not a multiplication)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the parentheses holding an input to a named function (not a multiplication)?

Worked Examples

Example 1

easy

f(x) = 3x^2 - 2x + 1

, find

f(-2)

Answer

f(-2) = 17

First step

Replace every

x

in the formula with

-2

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

Full solution

2
Evaluate: $f(-2) = 3(4) + 4 + 1 = 12 + 4 + 1$ .
3
$f(-2) = 17$ .

Function notation

f(x)

names the function (

f

) and its input variable (

x

). To evaluate

f(a)

, substitute

a

for every occurrence of

x

in the expression. Be careful with signs when substituting negative values — use parentheses around the substituted value.

Example 2

medium

g(x) = x^2 + 3x

, find and simplify

\frac{g(x+h) - g(x)}{h}

Example 3

medium

f(x) = 5x - 3

and

f(c) = 22

, find

c

Example 4

medium

f(x) = \sqrt{x+1}

, find the set of

x

for which

f

is defined.

Example 5

medium

f(x) = 2x - 1

and

g(x) = 3x + 4

, find

(f + g)(x)

and

(fg)(2)

Example 6

hard

f(x) = ax^2 + bx + c

and

f(0) = 4

f(1) = 9

f(-1) = 5

, find

a

b

c

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

f(x) = 2x - 1

and

g(x) = x^2 + 3

, find

(f \circ g)(x)

and

(g \circ f)(x)

Example 2

hard

A function satisfies

f(x+1) = 2f(x) + 3

for all

x

, with

f(0) = 1

. Find

f(1)

f(2)

, and

f(3)

Example 3

easy

f(x) = 2x + 3

, find

f(4)

Example 4

easy

g(x) = x^2 - 1

, find

g(3)

Example 5

easy

f(x) = 5

, find

f(100)

Example 6

easy

f(x) = 2x

, find

f(-3)

Example 7

easy

f(x) = x + 1

, find

f(a)

Example 8

easy

f(x) = x^2

, find

f(0)

Example 9

easy

h(t) = 3t - 2

, find

h(2)

Example 10

easy

f(x) = 4x - 1

and

f(c) = 7

, find

c

Example 11

medium

f(x) = x^2 + 2x

, find

f(x+1)

Example 12

medium

f(x) = 3x - 1

, find

f(a) + f(b)

where

a=2

b=5

Example 13

medium

f(x) = x^2

, compute

\frac{f(x+h) - f(x)}{h}

and simplify.

Example 14

medium

f(x) = 2x+1

and

g(x) = x^2

, find

f(g(2))

Example 15

medium

f(x) = \frac{1}{x-2}

, for what input is

f

undefined?

Example 16

medium

f(x) = x^2 - 4x

, find all

x

with

f(x) = 0

Example 17

medium

f(2x) = 4x + 1

, find a formula for

f(x)

Example 18

medium

f(x) = |x|

, find

f(-5) + f(5)

Example 19

medium

f(x) = 3x + 2

and

g(x) = x - 1

, find

g(f(1))

Example 20

challenge

f(x+1) = x^2 + 3

, find

f(x)

and then

f(5)

Example 21

challenge

f(x) = \frac{x}{x+1}

, simplify

f(f(x))

Example 22

challenge

f(x) = ax + b

f(1) = 5

, and

f(3) = 11

, find

a

and

b

Example 23

easy

f(x) = 7 - 2x

, find

f(3)

Example 24

easy

f(x) = 6x + 9

, find

f(0)

Example 25

easy

True or False:

f(x) = f \cdot x

when

f

is a function name.

Example 26

easy

f(x) = -x + 4

, find

f(-2)

Example 27

medium

f(x) = 2x + 1

and

g(x) = x - 4

, find

f(g(3))

Example 28

medium

f(x) = x^2 - 3x

, find

f(x-2)

and simplify.

Example 29

medium

f(x) = 3x + 2

, simplify

\frac{f(x+h) - f(x)}{h}

Example 30

medium

f(x) = \frac{x+3}{x-5}

, list all

x

where

f

is undefined.

Example 31

medium

f(x) = 4x - 1

, find

f(2x)

Example 32

medium

A function is given by the table

f(1)=4, f(2)=7, f(3)=10

. Find a linear formula consistent with the table.

Example 33

hard

f(x) = x^2 + 1

, find all

x

with

f(x) = f(2x)

Example 34

hard

f(x) = \frac{1}{x}

for

x \ne 0

, simplify

\frac{f(x+h) - f(x)}{h}

Example 35

hard

f(x) = 2x + 5

, find

f^{-1}(x)

(the inverse).

Example 36

hard

f(x) = 3x - 7

, find a function

g

such that

f(g(x)) = x

Example 37

hard

A piecewise function is

f(x) = \begin{cases} x + 1, & x < 0 \\ x^2, & x \ge 0 \end{cases}

. Find

f(-3) + f(2)

Example 38

challenge

A function satisfies

f(xy) = f(x) + f(y)

for all positive

x, y

and

f(2) = 1

. Find

f(8)

Example 39

challenge

f(x) + 2 f(1 - x) = x^2

for all

x

, find

f(2)

← Back to Function Notation Formula Explained Practice Mode

Related Concepts

Function Variables Evaluation

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitionvariablesevaluation