Function Examples in Math

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

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

Concept Recap

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.

A machine: put something in, get exactly one thing out. Same input always gives same output.

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: A function is a rule that pairs each allowed input with one and only one output.

Common stuck point: The procedure for function is the easy part; the trap is calling any equation in $x$ and $y$ a function. Asking "Does every allowed input give exactly one output, never two?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does every allowed input give exactly one output, never two?

Worked Examples

Example 1

easy

Determine whether the relation

\{(1, 3), (2, 5), (3, 7), (4, 9)\}

is a function.

Answer

\text{Yes, it is a function.}

First step

A relation is a function if every input (first element) maps to exactly one output (second element).

Full solution

2
List the inputs: $1, 2, 3, 4$ . Each input appears exactly once, so each has a unique output.
3
Since no input is repeated with a different output, this relation is a function.

A function assigns exactly one output to each input. The vertical line test for graphs and the uniqueness of first elements in ordered pairs are equivalent ways to check this.

Example 2

medium

Does the equation

x^2 + y^2 = 25

define

y

as a function of

x

Example 3

medium

Given

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

, find

f(4)

and determine if

g = \{(1,2), (3,4), (1,5)\}

is a function.

Example 4

medium

Decide whether

y^2 = x

defines

y

as a function of

x

Example 5

medium

f(x) = 2x + 3

and

f(a) = 11

, find

a

Example 6

hard

Given

f(x) = \dfrac{x + 1}{x - 1}

, find

f(f(2))

Example 7

hard

A graph passes the vertical line test but two horizontal lines each cut it in 3 places. Is it a function? Is it one-to-one?

Example 8

hard

Let

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

. Find

f(-3) + f(2)

Example 9

challenge

Find a function

f: \mathbb{R} \to \mathbb{R}

such that

f(f(x)) = x

for all

x

, other than

f(x) = x

Practice Problems

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

Example 1

easy

Is the relation

\{(2, 4), (3, 6), (2, 8)\}

a function? Explain.

Example 2

hard

A graph passes through the points

(1, 2)

(2, 5)

(3, 5)

, and

(1, 7)

. Does this graph represent a function? Explain using the vertical line test.

Example 3

easy

Does the rule 'add 3 to the input' define a function?

Example 4

easy

Is the set

\{(1,2),(2,3),(3,4)\}

a function?

Example 5

easy

Is the set

\{(1,2),(1,5),(3,4)\}

a function?

Example 6

easy

f(x)=2x+1

, find

f(4)

Example 7

easy

f(x)=x^2

, find

f(-3)

Example 8

easy

Can a function defined by a table assign output

7

to input

2

and also output

7

to input

5

Example 9

easy

Does

x^2+y^2=1

define

y

as a function of

x

Example 10

easy

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

, what is

f(0)

Example 11

medium

A function

f

satisfies

f(x)=3x-2

. For which input is

f(x)=10

Example 12

medium

f(x)=x^2-x

, compute

f(3)-f(2)

Example 13

medium

Given

f(x)=2x+1

and

f(a)=f(3)

, find

a

Example 14

medium

A function is defined by

f(x)=\begin{cases} x & x\ge 0 \\ -x & x<0 \end{cases}

. Find

f(-5)

Example 15

medium

f(x+1)=2x+5

, find

f(x)

Example 16

medium

Does the graph that passes through

(2,1)

and

(2,4)

represent a function?

Example 17

medium

f(x)=x^2

and

g(x)=x^2

but with domains

x\ge 0

and all reals, are

f

and

g

the same function?

Example 18

medium

f(x)=3x

and

f(f(x))=18

, find

x

Example 19

challenge

A function satisfies

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

for all

x

. Find

f(x)

Example 20

challenge

f

is defined on integers by

f(1)=1

and

f(n)=f(n-1)+n

. Find

f(5)

Example 21

challenge

f(x)=\frac{ax+b}{x}

and

f(1)=4

f(2)=3

, find

a+b

Example 22

medium

A function satisfies

f(2x)=4x+1

. Find

f(x)

Example 23

easy

Is the set

\{(0, 1), (1, 2), (2, 3), (3, 4)\}

a function?

Example 24

easy

f(x) = 3x - 5

, find

f(0)

Example 25

easy

g(x) = x^2 - 1

, find

g(-2)

Example 26

easy

Does

\{(1, 5), (2, 5), (3, 5), (4, 5)\}

define a function?

Example 27

medium

f(x) = \sqrt{x - 4}

, find

f(13)

Example 28

medium

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

, what value of

x

is not in the domain?

Example 29

medium

For

f(x) = x^2

, find

f(x+1) - f(x)

Example 30

medium

What is the domain of

f(x) = \sqrt{9 - x^2}

Example 31

medium

For

f(x) = |x|

, find

f(-7) + f(7)

Example 32

hard

State the domain of

f(x) = \dfrac{\sqrt{x+2}}{x - 3}

Example 33

hard

f(x) = ax + b

f(1) = 5

, and

f(3) = 11

, find

a

and

b

Example 34

hard

f(x) = x^2 - 4x + 1

, find

f(x + 2)

, simplified.

Example 35

hard

Find the range of

f(x) = x^2 + 4

for

x \in \mathbb{R}

Example 36

hard

For

f(x) = x^2

, simplify

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

assuming

h \ne 0

Example 37

challenge

How many functions

f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}

are one-to-one?

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Related Concepts

Domain Range Function Notation