Question

How do I solve and what would the answer be?

Answer 1

Given the function below

$\begin{gathered} f(x)=\frac{2}{x-5} \\ \therefore y=\frac{2}{x-5} \end{gathered}$

The inverse of the function is denoted by

$f^{-1}(x)_{}$

It can be obtained through the process below

$y=\frac{2}{x-5}$

Swap the position of y and x in the equation above

$x=\frac{2}{y-5}$

Solve for y in the resulting equation from the swap

$\begin{gathered} By\text{ cross multiplying} \\ x(y-5)=2 \\ \text{Divide both sides by x} \\ \frac{x(y-5)}{x}=\frac{2}{x} \end{gathered}$$\begin{gathered} y-5=\frac{2}{x} \\ \text{Add -5 to both sides} \\ y-5+5=\frac{2}{x}+5 \end{gathered}$$y=\frac{2}{x}+5$

Hence, the inverse of f(x) is

$f^{-1}(x)=\frac{2}{x}+5$