Q. 100 - Precalculus Enhanced with Graphing Utilities

Question

Give an example of a function whose domain is the set of real numbers and that is neither increasing nor decreasing on its domain, but is one-to-one.

Step-by-Step Solution

Answer

An example of a function is let $f\left(x\right)=k$ where $k$ is constant. Now, divide the domain of the function into piecewise like k  is constant for $1\le x\le 2$. Take another function $f\left(x\right)=a$ where a is constant for $2\le x\le 3$. Thus, we can divide the domain piecewise to make it one-to-one, and because the function's value is constant, it is neither increasing nor decreasing.

Step 1. Given Information

We have to give an example of a function where the domain is the set of all real numbers and which is neither increasing nor decreasing on its domain but is one-to-one.

Step 2. Giving an example

An example of a function is let $f\left(x\right)=k$ where k is constant. Now, divide the domain of the function into piecewise like k  is constant for $1\le x\le 2$ . Take another function $f\left(x\right)=a$ where a is constant for $2\le x\le 3$ . Thus, we can divide the domain piecewise to make it one-to-one, and because the function's value is constant, it is neither increasing nor decreasing.