Question

determine the inverse of h(x)=x/x-2

Answer 1

Given:

$h(x)=\frac{x}{x-2}$

To find the inverse of h(x):

Let the equation be,

$y=\frac{x}{x-2}$

Swap x and y, we get

$x=\frac{y}{y-2}$

On solving we get,

$\begin{gathered} \frac{y-2}{y}=\frac{1}{x} \\ 1-\frac{2}{y}=\frac{1}{x} \\ -\frac{2}{y}=\frac{1}{x}-1 \\ -\frac{2}{y}=\frac{1-x}{x} \\ \frac{2}{y}=-(\frac{1-x}{x})_{} \\ \frac{2}{y}=\frac{x-1}{x} \\ \frac{2}{y}=\frac{x-1}{x} \\ \frac{1}{y}=\frac{x-1}{2x} \\ y=\frac{2x}{x-1} \end{gathered}$

Hence, the inverse of the h(x) is,

$\frac{2x}{x-1}$