Chapter 0.5, Problem 59E

To determine

To prove: the given function $g\left(x\right)$ is one-to-one.

Answer to Problem 59E

The function $g\left(x\right)$ is one-to-one.

Explanation of Solution

Given:

The function $f$ is one-to-one.

  $g\left(x\right)=-f\left(x\right)$

Concept used:

A function $f\left(x\right)$ is one-to-one on a domain $D$ if $f\left(a\right)\ne f\left(b\right)$ whenever $a\ne b$ .

Calculation:

Consider,

  $\begin{array}{c}g\left(a\right)\ne g\left(b\right)\\ -f\left(a\right)\ne -f\left(b\right)\\ f\left(a\right)\ne f\left(b\right)\end{array}$

Now since $f$ is one-to-one, therefore

  $a\ne b$

Hence $g$ is one-to-one.