Practice Many-to-One Mapping in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
For f(n)=n mod 5, name two integers mapping to 1.

Example 2

medium
To invert the many-to-one f(x)=x2, what must you do first?

Example 3

hard
Is f(x)=x2 restricted to [−2,2] many-to-one?

Example 4

easy
For f(x)=x4, find both real inputs that map to 16.

Example 5

challenge
A function f:{1,2,3,4}→{a,b}. What is the minimum number of inputs that must share an output?

Example 6

hard
Is the relation defined by x2+y2=25 many-to-one, one-to-many, both, or neither (as a relation from x to y)?

Example 7

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

Example 8

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

Example 9

medium
For f(x)=x2 restricted to [0,∞), is it still many-to-one?

Example 10

easy
For f(x)=x2, find both inputs that map to 25.

Example 11

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

Example 12

easy
For f(x)=∣x∣, find both inputs with f(x)=7.

Example 13

easy
Show that f(x)=x2−4 is a many-to-one function by finding two distinct inputs that produce the same output.

Example 14

challenge
How many real x map to 0 under f(x)=x2(x−1)(x+2)?

Example 15

medium
Is f(x)=sin⁡x many-to-one on [0,2π]? Give two inputs with the same output.

Example 16

easy
In the table (1,5),(2,5),(3,7), which inputs share an output?

Example 17

challenge
Over the reals, how many inputs map to 5 under f(x)=x2 versus f(x)=x3? Explain the difference.

Example 18

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

Example 19

medium
To make f(x)=x2 invertible, name a domain restriction.

Example 20

hard
If g(x) is many-to-one, is the composition f(g(x)) necessarily many-to-one?
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