Many-to-One Mapping Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy

For

f(x) = \sin(x)

, find two values of

x

[0, 2\pi]

such that

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

Solution

1
Solve $\sin(x) = \frac{1}{2}$ on $[0, 2\pi]$ : primary solution $x = \frac{\pi}{6}$ .
2
Sine is positive in the second quadrant: $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$ . Both $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ give output $\frac{1}{2}$ .

Answer

x = \dfrac{\pi}{6}

and

x = \dfrac{5\pi}{6}

The sine function is many-to-one because it is periodic and not monotone over its full domain. This is precisely why we must restrict its domain to

[-\pi/2, \pi/2]

to define an inverse (arcsin).

About Many-to-One Mapping

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.

Learn more about Many-to-One Mapping →

More Many-to-One Mapping Examples

Example 1 easy

Show that [formula] is a many-to-one function by finding two distinct inputs that produce the same o

Example 2 medium

The floor function [formula] maps every real number to the greatest integer [formula]. Show it is ma

Example 4 medium

For [formula], find all [formula] such that [formula], and explain why [formula] has no inverse on [

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Related Concepts

Function One-to-One Mapping Restricted Domain