Chapter 5.2, Problem 87AYU

To determine

To find: The function $f\left(x\right)=\left|x\right|$ is not one to one. Find a suitable restriction on the domain of $f$ so that the new function that results is one–to–one. Then find the inverse of the new function.

Answer to Problem 87AYU

Solution:

$f\left(x\right)=\left|x\right|,x\ge 0$ is one–to–one function only when $f\left(x\right)=x,x\ge 0$.

Explanation of Solution

Given:

Function $f\left(x\right)=\left|x\right|$.

Calculation:

Given function $f\left(x\right)=\left|x\right|,x\ge 0$ has two different values for $x$ that is,

$f\left(x\right)=-x,x\ge 0$   -----(1)

$f\left(x\right)=x,x\ge 0$   -----(2)

From (1), the inverse of $f$ cannot be the same as for $x$ values and it is multiplied by minus sign with all inputs, whereas from (2), the inverse of $f$ can have the same $x$ values.

So Equation (2) is the one–to–one function based on different input values for $x$.