# How do I find the inverse function of f(x)=e^(3x)-4 ? And what is it's range ?

Apr 16, 2018

To find the inverse algebraically, switch the x and ys.

$x = {e}^{3 y} - 4$

$x + 4 = {e}^{3 y}$

$\ln \left(x + 4\right) = \ln \left({e}^{3 y}\right)$

$\ln \left(x + 4\right) = 3 y$

$y = {f}^{-} 1 \left(x\right) = \frac{1}{3} \ln \left(x + 4\right)$

Since the inverse of the function is the original function reflected over the line $y = x$, the domain of the original function becomes the range of the inverse and vice versa. Since $y = {e}^{3 x} - 4$ has a domain of all the real numbers, $y = \frac{1}{3} \ln \left(x + 4\right)$ will have a range of all the real numbers.

Hopefully this helps!