Many-to-One Mapping Math Example 4

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Example 4

medium
For g(x)=x2−6x+9=(x−3)2g(x) = x^2 - 6x + 9 = (x-3)^2, find all xx such that g(x)=16g(x) = 16, and explain why gg has no inverse on R\mathbb{R}.

Solution

  1. 1
    Solve (x−3)2=16⇒x−3=±4⇒x=7(x-3)^2 = 16 \Rightarrow x-3 = \pm 4 \Rightarrow x = 7 or x=−1x = -1.
  2. 2
    Two distinct inputs (77 and −1-1) give the same output (1616), confirming many-to-one. Since gg is not one-to-one, no inverse exists on all of R\mathbb{R}.

Answer

x=7x = 7 and x=−1x = -1; gg has no inverse on R\mathbb{R}
Many-to-one functions are not invertible because an inverse would need to map one output back to two inputs, which would make the 'inverse' a relation, not a function.

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 3 easy

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

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