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 β€” it is a valid function (each input still has exactly one output) but has no inverse.

Multiple students can have the same gradeβ€”many inputs, one output.

Showing a random 20 of 50 problems.

Example 1

easy
For f(n)=nβ€Šmodβ€Š5f(n) = n \bmod 5, name two integers mapping to 11.

Example 2

medium
To invert the many-to-one f(x)=x2f(x)=x^2, what must you do first?

Example 3

hard
Is f(x)=x2f(x) = x^2 restricted to [βˆ’2,2][-2, 2] many-to-one?

Example 4

easy
For f(x)=x4f(x) = x^4, find both real inputs that map to 1616.

Example 5

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

Example 6

hard
Is the relation defined by x2+y2=25x^2 + y^2 = 25 many-to-one, one-to-many, both, or neither (as a relation from xx to yy)?

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)=x2f(x)=x^2 restricted to [0,∞)[0,\infty), is it still many-to-one?

Example 10

easy
For f(x)=x2f(x) = x^2, find both inputs that map to 2525.

Example 11

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

Example 12

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

Example 13

easy
Show that f(x)=x2βˆ’4f(x) = x^2 - 4 is a many-to-one function by finding two distinct inputs that produce the same output.

Example 14

challenge
How many real xx map to 00 under f(x)=x2(xβˆ’1)(x+2)f(x) = x^2(x-1)(x+2)?

Example 15

medium
Is f(x)=sin⁑xf(x)=\sin x many-to-one on [0,2Ο€][0,2\pi]? Give two inputs with the same output.

Example 16

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

Example 17

challenge
Over the reals, how many inputs map to 55 under f(x)=x2f(x)=x^2 versus f(x)=x3f(x)=x^3? Explain the difference.

Example 18

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

Example 19

medium
To make f(x)=x2f(x) = x^2 invertible, name a domain restriction.

Example 20

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