Question

Indicate whether the two functions are equal. If the two functions are not equal, then give an element of the domain on which the two functions have different values.

$f:Z\to Z,$ where $f\left(x\right)=x^3$

$g:Z\to Z,$ where $g\left(x\right)={\left|x\right|}^3$

Answer 1

Writing the modular function

Given functions are

$f:Z\to Z,$ where $f\left(x\right)=x^3$

$g:Z\to Z,$ where $g\left(x\right)={\left|x\right|}^3$

The function $\left|x\right|$ can be written as

$$\begin{gathered} \left|x\right|=\left\{\begin{array}{ll}x& x\ge 0\\ -x& x<0\end{array}\right. \\ \Rightarrow {\left|x\right|}^3=\left\{\begin{array}{ll}{x}^3& x\ge 0\\ -x^3& x<0\end{array}\right. \end{gathered}$$

Therefore, the function $f$ is not equal to function $g$ in the domain $\left(-\infty ,0\right)$.