Chapter 5.2, Problem 63E

To determine

To show : That monotonic increasing and decreasing function are one-to- one.

Answer to Problem 63E

The possible explanation is shown below.

Explanation of Solution

Given information :

Monotonic increasing and decreasing function.

Calculation :

In monotonic increasing or decreasing function for any $x_1<x_2$ ,then $f\left(x_1\right)<f\left(x_2\right)$ , and for any $x_1>x_2$ , then $f\left(x_1\right)>f\left(x_2\right)$ .

For any $x_1\ne x_2$ implies,

  $\begin{array}{l}{x}_1<x_2\text{ or }{x}_1>x_2\\ f\left(x_1\right)<f\left(x_2\right)\text{ or }f\left(x_1\right)>f\left(x_2\right)\\ f\left(x_1\right)\ne f\left(x_2\right)\end{array}$

Therefore, $f$ is one-to-one.

Hence,

Monotonic increasing and decreasing functions are one-to-one.