Verify that the functions f and g, are inverses of each other by showing that f(g(x))=x and g(f(x))=x. f(x)=-7/x-4    g(x)=4x-7/xPLEASE SHOW ALL WORK!!!!!

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to remember how the composition of two function works, hence `f(g(x)) = (fog)(x).`

Substituting g(x) for x in the equation of f(x) yields:

`f(g(x)) = f((4x-7)/x) = -7/((4x-7)/x - 4)`

You need to bring the terms `(4x-7)/x`  and 4 to a common denominator such that:

`(4x-7)/x - 4 = (4x - 7 - 4x)/x = (-7)/x`

Hence `f(g(x)) = -7/(-7/x)`

You need to multiply -7 by the reversed denominator such that:

`f(g(x)) = -7*(-x/7) =`  x

You need to prove that g(f(x)) = x, hence, substituting `-7/(x-4)`  for f(x) in the equation of g(x) yields:

`g(f(x)) = g(-7/(x-4)) = (4(-7/(x-4)) - 7)/x`

`` `g(f(x)) = (-28 - 7x + 28)/x(x-4) = -7/(x-4) = f(x)`

Hence, since g(f(x)) yields f(x), the functions f(x) and g(x) are not inverse to each other.