Question

How do you find the inverse of `f(x) =e^(2x-1)`?

Answer 1

`f^(-1)(x) = 1/2(ln x + 1)` for `x > 0`

Explanation:

First, set plug `y` for `f(x)`:

`y = e^(2x-1)`

Then, exchange `y` and `x` in the equation:

`x = e^(2y-1)`

Now, try to solve the equation for `y` in terms of `x`.

To do so, we need to apply the logarithmic function `ln` to both sides of the equation. Please note that `x > 0` needs to hold so that the logarithm is well defined.

`ln (x) = ln(e^(2y-1))`
`<=> ln(x) = 2y-1`
`<=> ln(x) + 1 = 2y`
`<=> 1/2(ln(x) + 1) = y`

The only thing left to do is replacing `y` with `f^(-1)(x)`.
The inverse function is

`f^(-1)(x) = 1/2(ln x + 1)` for `x > 0`