Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = (x-8)/(x+7) g(x) = (-7x-8)/(x-1) I'm running out of time. I'd appreciate if someone could just help me with this.

Answer 1

`f(x)=(x-8)/(x+7) \\ g(x)=(-7x-8)/(x-1) \\ \\ f(g(x))=((-7x-8)/(x-1)-8)/((-7x-8)/(x-1)+7)=((-7x-8)/(x-1)-(8(x-1))/(x-1))/((-7x-8)/(x-1)+(7(x-1))/(x-1))=((-7x-8)/(x-1)-(8x-8)/(x-1))/((-7x-8)/(x-1)+(7x-7)/(x-1))=((-7x-8-8x+8)/(x-1))/((-7x-8+7x-7)/(x-1))= \\ =((-13x)/(x-1))/((-13)/(x-1))=(-13x)/((x-1)) * ((x-1))/(-13)=x \\ \Downarrow \\ f(g(x))=x \\ \checkmark`


`g(f(x))=(-7((x-8)/(x+7))-8)/((x-8)/(x+7)-1)=((-7(x-8))/(x+7)-(8(x+7))/(x+7))/((x-8)/(x+7)-(1(x+7))/(x+7))=((-7x+56)/(x+7)-(8x+56)/(x+7))/((x-8)/(x+7)-(x+7)/(x+7))= \\ =((-7x+56-8x-56)/(x+7))/((x-8-x-7)/(x+7))=((-15x)/(x+7))/((-15)/(x+7))=(-15x)/((x+7)) * ((x+7))/(-15)=x \\ \Downarrow \\ g(f(x))=x \\ \checkmark`