Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quantity negative five x minus nine divided by quantity x minus one.

Answer 1

We are given the functions f and g with the rules:


`\displaystyle{ f(x)= (x-9)/(x+5)`

and


`\displaystyle{ g(x)= (-5x-9)/(x-1)`.


First, we prove that f(g(x))=x:


`\displaystyle{ f(g(x))= (g(x)-9)/(g(x)+5)=((-5x-9)/(x-1)-9)/((-5x-9)/(x-1)+5)=((-5x-9)/(x-1)-9\cdot(x-1)/(x-1))/((-5x-9)/(x-1)+5\cdot(x-1)/(x-1))\\\\`


`=\displaystyle { ( (-5x-9-9x+9)/(x-1) )/((-5x-9+5x-5)/(x-1))= (-5x-9-9x+9)/(-5x-9+5x-5) = (-14x)/(-14)=x.`


Similarly, we prove that g(f(x))=x:


`\displaystyle{ g(f(x))= (-5f(x)-9)/(f(x)-1)= (-5\cdot (x-9)/(x+5)-9\cdot (x+5)/(x+5))/((x-9)/(x+5)-(x+5)/(x+5))= (-5(x-9)-9(x+5))/(x-9-(x+5))`


`=\displaystyle{ (-5x+45-9x-45)/(x-9-x-5)= (-14x)/(-14)=x.`