Question

If `y=e^x`,then `f^-1(x)` is?

`If y=e^x,then f^-1(x)?`

Answer 1

The answer is `=lnx`

Explanation:

I assumed that

`f(x)=e^x`

Let, `y=e^x`

The domain of `x` is `RR`

The range of `y` is `RR**^+`

`f(x)` and `f^-1(x)` are reflections in the line `y=x`

Then

`ln(y)=ln(e^x)`

`lny=x`

`x=lny`

Exchanging `x` and `y` in the last equation

`y=lnx`

So, the inverse is

`f^-1(x)=lnx`

Verification by performing the composition of the functions

`fof^-1(x)=f(f^-1(x))=f(lnx)=e^(lnx)=x`

graph{(y-x)(y-e^x)(x-e^y)=0 [-5.34, 8.705, -2.81, 4.21]}