Question

How do you show that `f(x)=(x+3)/(x-2)` and `g(x)=(2x+3)/(x-1)` are inverse functions algebraically and graphically?

Answer 1

If `f(x)` is the initial function, and `f^-1(x)` is the inverse function, then

`f(f^-1(x)) = x`

So here we go:

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

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

`f(g(x)) = (5x)/(x- 1) * (x -1)/(5)`

`f(g(x)) = (5x)/5`

`f(g(x)) = x`

This proves that `f(x)` and `g(x)` are inverses.

Hopefully this helps!