Question

How do you show that `f(x)=(x-1)/(x+5)` and `g(x)=-(5x+1)/(x-1)` are inverse functions algebraically and graphically?

Answer 1

See below.

Explanation:

Two functions `f(x)` and `g(x)` are inverses of each other if and only if `f(g(x)) = g(f(x)) = x`.

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

`=(-(5x+1)/(x-1) - (x-1)/(x-1))/(-(5x+1)/(x-1) + (5(x-1))/(x-1))`

`=((-5x-1-x+1)/(x-1))/((-5x-1+5x-5)/(x-1))`

`=(-5x-1-x+1)/(-5x-1+5x-5)`

`=(-6x)/-6`

`=x`

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

`=-((5x-5)/(x+5) + (x+5)/(x+5))/((x-1)/(x+5)-(x+5)/(x+5))`

`=-((5x-5+x+5)/(x+5))/((x-1-x-5)/(x+5))`

`=-(5x-5+x+5)/(x-1-x-5)`

`=-(6x)/(-6)`

`=x`

Since we have proved `f(g(x))=x` and `g(f(x))=x`, we can say `f(g(x))=g(f(x))=x`.

`color(white)I`

In the graph below, `f(x)` is shown in red and `g(x)` in blue. If two functions `f(x)` and `g(x)` are inverses, then they will be reflections of each other over the line `y=x` (shown in green).

It's clear to see that `f(x)` and `g(x)` are reflections of each other over the line!