Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = quantity x minus eight divided by quantity x plus seven and g(x) = quantity negative seven x minus eight divided by quantity x minus one.

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 eight divided by quantity x plus seven and g(x) = quantity negative seven x minus eight divided by quantity x minus one.

mathstudent55 · Answer

$f(x) = \dfrac{x - 8}{x + 7} $

$g (x) = \dfrac{-7x - 8}{x - 1} $

$f(g(x)) = \dfrac{\frac{-7x - 8}{x - 1} - 8}{\frac{-7x - 8}{x - 1}  + 7} $

$~~~~~~~~~~~ = \dfrac{(x - 1)(\frac{-7x - 8}{x - 1} - 8)}{(x - 1)(\frac{-7x - 8}{x - 1}  + 7)} $

$~~~~~~~~~~~ = \dfrac{-7x - 8 - 8(x - 1)}{-7x - 8 + 7(x - 1)} $

$~~~~~~~~~~~ = \dfrac{-7x - 8 - 8x + 8}{-7x - 8 + 7x - 7} $

$~~~~~~~~~~~ = \dfrac{-15x}{-15} $

$~~~~~~~~~~~ = x$

mathstudent55 · Answer

That is how you show f(g(x)) = x.
Now do the same for g(f(x)).