Pre-calc/trig

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 and g(x) = -7x-8/x-1

Answer 1

Note: I'm guessing you were missing some parentheses in your f(x) and g(x)?

 

First show that f(g(x))=x. That means plug g(x) into f(x).

f(g(x)) = [g(x)-8] / [g(x)+7]

  = [(-7x-8)/(x-1) - 8] / [(-7x-8)/(x-1) + 7]

Simplifying by creating a common denominator, you get:

  = [(-7x-8)/(x-1) - 8(x-1)/(x-1)] / [(-7x-8)/(x-1) + 7(x-1)/(x-1)]

Now you can cancel out the common denominator (x-1) from top and bottom, and get:

  = (-7x-8-8(x-1)) / (-7x-8+7(x-1))

And simplifying gives:

  = (-7x-8-8x+8) / (-7x-8+7x-7)

  = -15x / -15

  = x

 

Then do the same thing for g(f(x)), which should also simplify to x, if f and g are indeed inverses of each other.