Many-to-One Mapping Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Many-to-One Mapping.

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 many-to-one function maps multiple distinct inputs to the same output — it is a valid function (each input still has exactly one output) but has no inverse.

Multiple students can have the same grade—many inputs, one 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 many-to-one function is a valid function where different inputs can land on the same output, so it has no inverse.

Common stuck point: The procedure for many-to-one mapping is the easy part; the trap is calling a many-to-one function 'not a function'. Asking "Do two or more distinct inputs produce the same output?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do two or more distinct inputs produce the same output?

Worked Examples

Example 1

easy

Show that

f(x) = x^2 - 4

is a many-to-one function by finding two distinct inputs that produce the same output.

Answer

f(3)=f(-3)=5

;

f

is many-to-one

First step

Try

x = 3

f(3) = 9-4 = 5

. Try

x = -3

f(-3) = 9-4 = 5

Full solution

2
We have $f(3) = f(-3) = 5$ but $3 \neq -3$ . This confirms many-to-one behavior.
3
This occurs for all pairs $\pm x$ (except $x=0$ ) because squaring removes the sign.

A many-to-one function maps multiple distinct inputs to the same output. Even functions (

f(-x)=f(x)

) are inherently many-to-one because symmetric pairs of inputs are mapped to identical values.

Example 2

medium

The floor function

f(x) = \lfloor x \rfloor

maps every real number to the greatest integer

\leq x

. Show it is many-to-one and find

f^{-1}(\{3\})

Example 3

medium

For

f(x) = x^2 + 1

, find all

x

with

f(x) = 5

Example 4

hard

A function

f: \{1,2,3,4,5\} \to \{a,b,c\}

. Use pigeonhole to find the minimum number of inputs that must share an output.

Practice Problems

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

Example 1

easy

For

f(x) = \sin(x)

, find two values of

x

[0, 2\pi]

such that

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

Example 2

medium

For

g(x) = x^2 - 6x + 9 = (x-3)^2

, find all

x

such that

g(x) = 16

, and explain why

g

has no inverse on

\mathbb{R}

Example 3

easy

Does

f(x)=x^2

map more than one input to

4

Example 4

easy

Is a many-to-one function still a valid function?

Example 5

easy

In the table

(1,5),(2,5),(3,7)

, which inputs share an output?

Example 6

easy

Does a many-to-one function have a simple inverse?

Example 7

easy

Is the map 'student to letter grade' many-to-one in a class of 30?

Example 8

easy

Is a function ever allowed to be one-to-many?

Example 9

easy

For

f(x)=|x|

, name two inputs giving output

3

Example 10

easy

Does a constant function

f(x)=2

qualify as many-to-one?

Example 11

medium

For

f(x)=x^2

restricted to

[0,\infty)

, is it still many-to-one?

Example 12

medium

How many inputs map to

9

under

f(x)=x^2

on the reals?

Example 13

medium

Under

f(x)=x^2-1

, find all inputs mapping to

0

Example 14

medium

f(x)=\sin x

many-to-one on

[0,2\pi]

? Give two inputs with the same output.

Example 15

medium

A function rounds any real to the nearest integer. Is it many-to-one?

Example 16

medium

Under

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

, how many inputs give output

4

Example 17

medium

Does the map 'integer to its remainder mod 3' send many inputs to

0

Example 18

medium

To invert the many-to-one

f(x)=x^2

, what must you do first?

Example 19

challenge

For

f(x)=x^2-4x

, find all inputs mapping to

-3

and explain why there are two.

Example 20

challenge

Over the reals, how many inputs map to

5

under

f(x)=x^2

versus

f(x)=x^3

? Explain the difference.

Example 21

challenge

A function

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

. What is the minimum number of inputs that must share an output?

Example 22

medium

Under

f(x)=x^2

, list the inputs mapping to

1

Example 23

easy

For

f(x) = x^2

, find both inputs that map to

25

Example 24

easy

For the floor function

\lfloor x \rfloor

, find two real inputs that both map to

2

Example 25

easy

For

f(x) = \cos(x)

, find two real

x

with

f(x) = 1

Example 26

easy

For

f(x) = |x|

, find both inputs with

f(x) = 7

Example 27

easy

For

f(x) = x^4

, find both real inputs that map to

16

Example 28

easy

For

f(n) = n \bmod 5

, name two integers mapping to

1

Example 29

medium

For

f(x) = x^2 - 2x

, find all

x

with

f(x) = 0

Example 30

medium

For

f(x) = \sin(x)

[0, 2\pi]

, list all

x

with

f(x) = \frac{\sqrt{2}}{2}

Example 31

medium

For

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

, find all

x

with

f(x) = 9

Example 32

medium

Is the clock-hour function

f(t) = t \bmod 12

many-to-one?

Example 33

medium

f(x) = x^3 + x

many-to-one over

\mathbb{R}

Example 34

medium

To make

f(x) = x^2

invertible, name a domain restriction.

Example 35

medium

For

f(x) = |x - 3|

, find all

x

with

f(x) = 2

Example 36

medium

How many real

x

satisfy

\sin(x) = 0

[0, 2\pi]

Example 37

hard

For

f(x) = x^2 - 6x

, find all

x

with

f(x) = 7

Example 38

hard

How many real

x

map to

4

under

f(x) = x^4

Example 39

hard

f(x) = x^2

restricted to

[-2, 2]

many-to-one?

Example 40

hard

For

f(x) = \lceil x \rceil

(ceiling), how many real

x

map to

3

Example 41

hard

Is the relation defined by

x^2 + y^2 = 25

many-to-one, one-to-many, both, or neither (as a relation from

x

y

Example 42

hard

g(x)

is many-to-one, is the composition

f(g(x))

necessarily many-to-one?

Example 43

challenge

How many real

x

map to

0

under

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

Example 44

challenge

A function

h:\mathbb{Z} \to \{0,1\}

defined by

h(n) = n \bmod 2

. How many integers map to

0

← Back to Many-to-One Mapping Formula Explained Practice Mode

Related Concepts

Function One-to-One Mapping Restricted Domain

Background Knowledge

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

function definition