Question

How do you find the inverse of `e^-x` and is it a function?

Answer 1

`-ln(x)`

Explanation:

We have the function

`y=e^-x`

To find its inverse, swap `y` and `x`.

`x=e^-y`

Solve for `y` by taking the natural logarithm of both sides.

`ln(x)=-y`

`y=-ln(x)`

This is a function. We knew that it would be because of the graph of `y=e^-x`:

graph{e^-x [-16.36, 34.96, -5.25, 20.4]}

There is only one `x` value for every `y` value, so its inverse will only have one `y` value for every `x` value, the definition of a function, i.e., the inverse will pass the vertical line test:

graph{-lnx [-9.77, 55.18, -7.23, 25.23]}