Many-to-One Mapping Math Example 1

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

Example 1

easy
Show that f(x)=x2โˆ’4f(x) = x^2 - 4 is a many-to-one function by finding two distinct inputs that produce the same output.

Solution

  1. 1
    Try x=3x = 3: f(3)=9โˆ’4=5f(3) = 9-4 = 5. Try x=โˆ’3x = -3: f(โˆ’3)=9โˆ’4=5f(-3) = 9-4 = 5.
  2. 2
    We have f(3)=f(โˆ’3)=5f(3) = f(-3) = 5 but 3โ‰ โˆ’33 \neq -3. This confirms many-to-one behavior.
  3. 3
    This occurs for all pairs ยฑx\pm x (except x=0x=0) because squaring removes the sign.

Answer

f(3)=f(โˆ’3)=5f(3)=f(-3)=5; ff is many-to-one
A many-to-one function maps multiple distinct inputs to the same output. Even functions (f(โˆ’x)=f(x)f(-x)=f(x)) are inherently many-to-one because symmetric pairs of inputs are mapped to identical values.

About Many-to-One Mapping

A many-to-one function maps multiple distinct inputs to the same output โ€” it is a valid function (each input still has exactly one output) but has no inverse.

Learn more about Many-to-One Mapping โ†’

More Many-to-One Mapping Examples

Example 2 medium

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

Example 3 easy

For [formula], find two values of [formula] in [formula] such that [formula].

Example 4 medium

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

โ† All Many-to-One Mapping Examples Many-to-One Mapping Concept