Question

What is the inverse function of `y=5^x+1`?

Answer 1

`f(y) = log_5(y-1)`
`f^(-1)(x) = log_5(y-1)`
`x = log_5(y-1)`

Explanation:

Isolate the terms with an `x`

`y = 5^x + 1`
`y - 1 = 5^x`

Take the log of base 5 of both sides

`log_5(y-1) = x`

Other common notations will have you replace `x` for `f(y)` or replace `x` for `f^(-1)(x)` and `y` for `x`, like this:

`f(y) = log_5(y-1)`
`f^(-1)(x) = log_5(y-1)`
`x = log_5(y-1)`

I personally care more for the first and the third, but they all have their strengths. The first cleary indicates this is a function in terms of `y`, the second that this is an inverse function to find `x` and the third explicits the relation between `x` and `y`.

Your teacher will probably want you to put down the second or the third though, so try to remember that step.