One-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 One-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 one-to-one (injective) function maps every distinct input to a distinct output — no two different inputs produce the same output.

No two inputs share the same output—like social security numbers.

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 one-to-one function never lets two distinct inputs share the same output.

Common stuck point: The procedure for one-to-one mapping is the easy part; the trap is confusing one-to-one with being a function. Asking "Does every output value come from at most one input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does every output value come from at most one input?

Worked Examples

Example 1

easy

Determine whether

f(x) = 2x + 3

is one-to-one by (a) the definition and (b) the horizontal line test.

Answer

f(x)=2x+3

is one-to-one

First step

Definition: assume

f(a)=f(b)

. Then

2a+3=2b+3 \Rightarrow 2a=2b \Rightarrow a=b

. So

f(a)=f(b) \Rightarrow a=b

f

is one-to-one. ✓

Full solution

2
Horizontal line test: $f(x)=2x+3$ is a line with positive slope. Any horizontal line $y=c$ intersects it at exactly one point $(x = (c-3)/2)$ .
3
Both methods confirm $f$ is one-to-one (injective).

A one-to-one function has no two distinct inputs sharing an output:

f(a)=f(b)\Rightarrow a=b

. Linear functions with non-zero slope are always one-to-one because they are strictly monotone.

Example 2

medium

Show that

g(x) = x^3

is one-to-one on

\mathbb{R}

, then find its inverse function.

Example 3

medium

Show that

f(x) = 5 - 2x

is one-to-one and find its inverse.

Example 4

medium

Find the inverse of

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

(with

x \ge -4

) and state its domain.

Example 5

medium

Determine whether the floor function

f(x) = \lfloor x \rfloor

is one-to-one on

\mathbb{R}

Example 6

hard

Find the inverse of

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

and state its domain.

Example 7

hard

A function

f:\mathbb{Z} \to \mathbb{Z}

is defined by

f(n) = 2n + 5

. Show it is one-to-one but not onto.

Example 8

hard

Show that

f(x) = e^x + x

is one-to-one on

\mathbb{R}

Example 9

challenge

Find all real

b

such that

f(x) = x^3 + bx

is one-to-one on

\mathbb{R}

Practice Problems

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

Example 1

easy

Which functions are one-to-one? (A)

f(x)=x^2

\mathbb{R}

. (B)

f(x)=e^x

. (C)

f(x)=|x|

Example 2

hard

Find the inverse of

h(x) = \dfrac{2x+1}{x-3}

and state its domain.

Example 3

easy

f(x)=x+5

one-to-one? (Do different inputs give different outputs?)

Example 4

easy

Apply the horizontal line test: a line

y=2x

. Is it one-to-one?

Example 5

easy

f(x)=x^2

one-to-one over all reals?

Example 6

easy

Does the table

(1,2),(2,4),(3,6)

define a one-to-one function?

Example 7

easy

Does the table

(1,5),(2,5),(3,9)

define a one-to-one function?

Example 8

easy

f(x)=3x-1

one-to-one?

Example 9

easy

A constant function

f(x)=7

. Is it one-to-one?

Example 10

easy

Is the map 'each person to their fingerprint' one-to-one?

Example 11

medium

f(x)=x^3

one-to-one over the reals? Justify.

Example 12

medium

Restrict the domain of

f(x)=x^2

to make it one-to-one. Give a valid restriction.

Example 13

medium

A graph dips down then rises (a parabola). State why it fails the horizontal line test.

Example 14

medium

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

(for

x\neq0

) one-to-one?

Example 15

medium

How many distinct inputs can map to a single output in a one-to-one function?

Example 16

medium

f(x)=|x|

one-to-one on the reals? If not, restrict it.

Example 17

medium

f

is one-to-one and

f(a)=f(b)

, what can you conclude?

Example 18

medium

Is the doubling map on integers,

n\mapsto 2n

, one-to-one?

Example 19

challenge

Determine all real

k

for which

f(x)=kx+3

is one-to-one.

Example 20

challenge

f(x)=x^2-4x+7

one-to-one on the reals? Find the largest interval

[a,\infty)

where it is.

Example 21

challenge

A function maps

\{1,2,3\}

\{a,b\}

. Can it be one-to-one? Explain.

Example 22

medium

Is the map 'day of week to its number

1..7

' one-to-one?

Example 23

easy

f(x) = 7x

one-to-one on

\mathbb{R}

Example 24

easy

Is the function given by the table

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

one-to-one?

Example 25

easy

Which function is NOT one-to-one? (A)

f(x)=2x-1

(B)

f(x)=x^4

\mathbb{R}

(C)

f(x)=x^5

Example 26

easy

f(x) = \ln(x)

(0,\infty)

one-to-one?

Example 27

medium

Restrict the domain of

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

so that it becomes one-to-one. State a valid restriction.

Example 28

medium

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

one-to-one on its domain?

Example 29

medium

f(x) = \sin x

one-to-one on

\mathbb{R}

? If not, give a domain restriction that makes it one-to-one.

Example 30

medium

f(x) = 2^x

one-to-one?

Example 31

medium

Suppose

f

and

g

are both one-to-one. Is

f \circ g

one-to-one?

Example 32

medium

Which of these is one-to-one on

\mathbb{R}

? (A)

f(x) = \cos x

(B)

f(x) = x^3 + x

(C)

f(x) = x^2 + 1

Example 33

hard

f(x) = x^3 - 3x + 4

one-to-one on

\mathbb{R}

? Justify using the derivative.

Example 34

hard

For what values of

a

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

one-to-one on

[0, \infty)

Example 35

hard

f(x) = x|x|

one-to-one on

\mathbb{R}

Example 36

hard

How many one-to-one functions are there from a 3-element set to a 5-element set?

Example 37

challenge

Let

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

satisfy

f(f(x)) = x

for all

x

. Must

f

be one-to-one?

Example 38

challenge

Let

f:\mathbb{N} \to \mathbb{N}

be one-to-one. Must its image be all of

\mathbb{N}

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Related Concepts

Function Inverse Function Horizontal Line Test

Background Knowledge

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

function definition