# How do you find the inverse of g(x)= (x + 2) / (x - 3)?

Jun 29, 2015

Find the inverse of $g \left(x\right)$ by simplifying it to have one occurrence of $x$ then rearranging to express $x$ in terms of $g \left(x\right)$

${g}^{-} 1 \left(y\right) = 3 + \frac{5}{y - 1}$

#### Explanation:

Let $y = g \left(x\right)$

Then

$y = \frac{x + 2}{x - 3} = \frac{x - 3 + 5}{x - 3} = 1 + \frac{5}{x - 3}$

Subtract $1$ from both ends to get:

$y - 1 = \frac{5}{x - 3}$

Multiply both sides by $\left(x - 3\right)$ to get:

$\left(y - 1\right) \left(x - 3\right) = 5$

Divide both sides by $\left(y - 1\right)$ to get:

$x - 3 = \frac{5}{y - 1}$

Add $3$ to both sides to get:

$x = 3 + \frac{5}{y - 1}$

So ${g}^{-} 1 \left(y\right) = 3 + \frac{5}{y - 1}$