Question

If $f\left(x\right)$ is defined in $\left[-3,2\right]$, then find the domain of definition of $f\left(\left[|x|\right]\right)$ and $f\left(\left[2x+3\right]\right)$

Answer 1

Domain of $f\left(x\right)$ of $\left[-3,2\right]$
For domain of $f\left(\left[|x|\right]\right)$
because $\left[\left|x\right|\right]$ is always positive
$0\le \left[\left|x\right|\right]\le 2$
$0<\left|x\right|<3$
$-3<x<3$
For domain of $f\left[\left(2x+3\right)\right]$
$\begin{array}{c}-3\le \left[2x+3\right]\le 3\\ -3\le \left(2x+3\right)<3\\ -6\le 2x<0\\ -3\le x<0\end{array}$