Practice Many-to-One Mapping in Math

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

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.

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.

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

easy
For f(x) = \sin(x), find two values of x in [0, 2\pi] such that f(x) = \frac{1}{2}.

Example 4

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}.
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