# How to find the equation of the inverse function (the one below)?

## $f \left(x\right) = \frac{1}{- \frac{1}{3} \left(x - 3\right)} - 4$

Feb 14, 2017

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

#### Explanation:

Let:

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

and solve for $x$ in terms of $y$.

First add $4$ to both ends to get:

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

Take the reciprocal of both sides of the equation to get:

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

Multiply both sides by $- 3$ to get:

$- \frac{3}{y + 4} = x - 3$

Add $3$ to both sides and transpose to get:

$x = 3 - \frac{3}{y + 4}$

So the inverse function can be written as:

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