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

## What is the inverse of $y = - {e}^{-} x$?

Dec 19, 2017

#### Explanation:

The inverse function of $f \left(x\right) = - {e}^{-} x$ is found by solving for $x$

$y = - {e}^{-} x$

$- y = {e}^{-} x$

$- x = \ln \left(- y\right)$

$x = - \ln \left(- y\right)$

So ${f}^{-} 1 \left(x\right) = - \ln \left(- x\right)$ $\text{ }$ (Domain is $x < 0$)

Note that the compositions are the composition identities $i d \left(x\right) = x$ on the domains.

$f \left({f}^{-} 1 \left(x\right)\right) = x$ and ${f}^{-} 1 \left(f \left(x\right)\right) = x$

The additive inverse is ${e}^{-} x$ because $\left(- {e}^{-} x\right) + {e}^{-} x = 0$ (the additive identity)

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