Question

How do you find the inverse of `f(x)=(e^x)+1`?

Answer 1

If `f^(-1)(x)` is the inverse of `f(x)=(e^x)+1` then
`color(white)("XXX")f^(-1)(x)=ln(x-1)`
(with some obvious limitations since `ln(x-1)` is not defined for `x<=1`)

Explanation:

By definition of inverse
`color(white)("XXX")f(f^(-1)(x)) = x`

and since `f(x)=(e^x)+1`

`color(white)("XXX")f(f^-1)(x)) = e^(f^(-1)(x))+1`

and therfore
`color(white)("XXX")e^(f^(-1)(x))+1 = x`

`color(white)("XXX")e^(f^(-1)(x) = x-1`

Taking the natural log of both sides:
`color(white)("XXX")f^(-1)(x) = ln(x-1)`