Find `fog` and `gof` , if `f(x)=e^x` , `g(x)=(log)_e x`

Explanation

`f(x)=e^{x}, g(x)=ln x`
`
f circ g(x)=e^{ln x}=x ; g circ f(x)=ln (e^{x})=x`

Hence,` f circ g(x)=g circ f(x)` proved
`
Let f(x)=y ; g(x)=y`
`
e^{x}=y ; y=ln x`
`
x=ln y ; x=e^{y}`

therefore `f^{-1}(x)=ln x ; f^{-1}(x)=e^{x}`