Question

Inverse of f(x)=(x-3)/(x+2)

Answer 1

`f^(-1)(x)=(2x+3)/(1-x)`

Explanation:

Find the inverse of the given function, f(x).

`f(x)=(x-3)/(x+2)`

(1) - Switch x and f(x)

`f(x)=(x-3)/(x+2)\\\\\Longrightarrow x=(f(x)-3)/(f(x)+2)`

(2) - Solve for f(x) using algebraic techniques

`x=(f(x)-3)/(f(x)+2)\\ \\\\\Longrightarrow (f(x)+2)x=f(x)-3\\ \\\\\Longrightarrow xf(x)+2x=f(x)-3\\ \\\\\Longrightarrow 2x=f(x)-3-xf(x)\\ \\ \\\Longrightarrow 2x+3=f(x)-xf(x) \\\\\\\Longrightarrow 2x+3=f(x)(1-x)\\ \\\\\therefore f(x)=(2x+3)/(1-x)`

(3) - Replace f(x) with f^-1(x)

`f(x)=(2x+3)/(1-x) \\\\\Longrightarrow \boxed{\boxed{f^(-1)(x)=(2x+3)/(1-x) }}`

Therefore, the inverse is found.