Question

If function $f:R\to R$ and $g:R\to R$ are given by $f\left(x\right)=|x|$ and $g\left(x\right)=\left[x\right]$, (where $\left[x\right]$ is greatest integer function) find $f\circ g(-\frac{1}{2})$ and $g\circ f(-\frac{1}{2})$.

Answer 1

Given $f\left(x\right)=\left|x\right|,g\left(x\right)=\left[x\right]$

$f\circ g\left(x\right)=\left|\left[x\right]\right|$

$[-\frac{1}{2}]=-1+\frac{1}{2}$

$g\circ f\left(x\right)=\left[\left|x\right|\right]$

$f\circ g(-\frac{1}{2})=\left|[-\frac{1}{2}]\right|=|-1|=1$

$g\circ f(-\frac{1}{2})=\left[\frac{1}{2}\right]=[0+\frac{1}{2}]=0$