Answer:
[tex]f^{-1}[/tex](x) = [tex]\frac{2x}{x-1}[/tex]
Step-by-step explanation:
let y = f(x)
then y = [tex]\frac{x}{x-2}[/tex]
rearrange making x the subject, cross- multiply
y(x - 2) = x
xy - 2y = x ( subtract x from both sides )
xy - x - 2y = 0 ( add 2y to both sides )
xy - x = 2y ( factor out x from each term on the left side )
x(y - 1) = 2y ( divide both sides by (y - 1)
x = [tex]\frac{2y}{y-1}[/tex]
change y back into terms of x
[tex]f^{-1}[/tex](x) = [tex]\frac{2x}{x-1}[/tex]
