Question

What is the inverse of `y= e^(x-1)-1` ?

Answer 1

`f^(-1)(x) = ln(x+1) +1`

Explanation:

To compute the inverse, you need to follow the following steps:

1) swap `y` and `x` in your equation:

`x = e^(y-1) - 1`

2) solve the equation for `y`:

... add `1` on both sides of the equation...

`x + 1 = e^(y-1)`

... remember that `ln x` is the inverse function for `e^x` which means that both `ln(e^x) = x` and `e^(ln x) = x` hold.
This means that you can apply `ln()` on both sides of the equation to "get rid" of the exponential function:

`ln(x+1) = ln(e^(y-1))`

`ln(x+1) = y-1`

... add `1` on both sides of the equation again...

`ln(x+1) + 1 = y`

3) Now, just replace `y` with `f^(-1)(x)` and you have the result!

So, for

`f(x) = e^(x-1) - 1`,

the inverse function is

`f^(-1)(x) = ln(x+1) +1`

Hope that this helped!