Question

If $f\left(x\right)=e^x$ and $g\left(x\right)={\log }_{e}x$, then show that $fog=gof$ and find $f^{-1}$ and $g^{-1}$

Answer 1

$f\left(x\right)=e^x;g\left(x\right)=\ln x$

$f\circ g\left(x\right)=e^{\ln x}=x;g\circ f\left(x\right)=\ln \left(e^x\right)=x$

Hence, $f\circ g\left(x\right)=g\circ f\left(x\right)$ proved

Let $f\left(x\right)=y;g\left(x\right)=y$

$e^x=y;y=\ln x$

$x=\ln y;x=e^y$

$\therefore f^{-1}\left(x\right)=\ln x;f^{-1}\left(x\right)=e^x$