# What is the inverse of f(x) = 3^(x^2-3x) ?

Nov 24, 2015

$y = \frac{3}{2} \pm \sqrt{{\log}_{3} x + \frac{9}{4}}$

#### Explanation:

$y = {3}^{{x}^{2} - 3 x}$

Flip $x$ and $y$.

$x = {3}^{{y}^{2} - 3 y}$

Solve for $y$.

${\log}_{3} x = {\log}_{3} \left({3}^{{y}^{2} - 3 y}\right)$

${\log}_{3} x = {y}^{2} - 3 y$

${\log}_{3} x + \frac{9}{4} = {y}^{2} - 3 y + \frac{9}{4}$

${\log}_{3} x + \frac{9}{4} = {\left(y - \frac{3}{2}\right)}^{2}$

$\pm \sqrt{{\log}_{3} x + \frac{9}{4}} = y - \frac{3}{2}$

$y = \frac{3}{2} \pm \sqrt{{\log}_{3} x + \frac{9}{4}}$