Mathematics - Functions Question with Solution | TestHub

Given: $f\left(x\right)=\left\{\begin{array}{cc}{\log }_{e}x,& x>0\\ e^{-x},& x\le 0\end{array}\right.$ and $g\left(x\right)=\left\{\begin{array}{cc}x,& x\ge 0\\ e^x,& x<0\end{array}\right.$

$\Rightarrow g\left(f\left(x\right)\right)=\left\{\begin{array}{cc}f\left(x\right),& f\left(x\right)\ge 0\\ e^{f\left(x\right)},& f\left(x\right)<0\end{array}\right.$

$\Rightarrow g\left(f\left(x\right)\right)=\left\{\begin{array}{cc}{e}^{-x},& \left(-\infty ,0\right]\\ e^{\log x},& \left(0,1\right)\\ \log x,& \left[1,\infty \right)\end{array}\right.$

Graph of $g\left(f\left(x\right)\right)$ is given below:

Range of $g\left(f\left(x\right)\right)=[0,\infty )\ne$Codomain

So, $g\left(f\left(x\right)\right)$ is many-one and into.