Chapter 1.4, Problem 90E

To determine

Find the reason of difference in domains of f and g .

Answer to Problem 90E

The reasons is that, $f\left(x\right)$ is an even root function and hence it cannot be negative and $g\left(x\right)$ is an odd root function and thus can have negative values underneath the cube root sign.So,both the functions have different domain

Explanation of Solution

Given:

The functions

  $\begin{array}{l}f\left(x\right)=\sqrt{x–2}\hfill \\ g\left(x\right)=\sqrt{x–2}\hfill \end{array}$

Consider the following functions for assessing their domain,

  $\begin{array}{l}f\left(x\right)=\sqrt{x–2}\hfill \\ g\left(x\right)=\sqrt{x–2}\hfill \end{array}$

  $f\left(x\right)$ is an even root function and hence it cannot be negative. This restricts $x\ge 2$ . Thus its domain is,

  $x\in \left[2,\infty \right)$

  $g\left(x\right)$ is an odd root function and thus can have negative values underneath the cube root sign.

Therefore, $x$ can have any real value on the number line. Hence its domain is,

  $x\in \left(–\infty ,\infty \right)$

So, both the functions have different domain.