Question

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

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

Answer 1

Please see below.

Explanation:

The inverse function of `f(x) = -e^-x` is found by solving for `x`

`y = -e^-x`

`-y=e^-x`

`-x=ln(-y)`

`x = -ln(-y)`

So `f^-1(x) = -ln(-x)` `" "` (Domain is `x < 0`)

Note that the compositions are the composition identities `id(x) = x` on the domains.

`f(f^-1(x)) = x` and `f^-1(f(x)) = x`

The additive inverse is `e^-x` because `(-e^-x)+ e^-x = 0` (the additive identity)

The multiplicative inverse is `-e^x` because (-e^-x) * -e^x = 1# (the multiplicative identity)