SOLUTION: I need help on showing that functions are inverse. F(x)=10+(8/x) and g(x)=8/(x-10)

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Question 187772: I need help on showing that functions are inverse.
F(x)=10+(8/x) and g(x)=8/(x-10)
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
In order to show that f%28x%29 and g%28x%29 are inverses of one another, you need to show that both f%28g%28x%29%29=x and g%28f%28x%29%29=x.



Show that f%28g%28x%29%29=x:


f%28x%29=10%2B8%2Fx Start with the first function.


f%28g%28x%29%29=10%2B8%2F%288%2F%28x-10%29%29 Plug in g%28x%29=8%2F%28x-10%29


f%28g%28x%29%29=10%2B8%2A%28%28x-10%29%2F8%29 Flip the second lower fraction and multiply.


f%28g%28x%29%29=10%2Bcross%288%29%2A%28%28x-10%29%2Fcross%288%29%29 Cancel out the common terms.


f%28g%28x%29%29=10%2Bx-10 Simplify.


f%28g%28x%29%29=x Combine like terms.


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Show that g%28f%28x%29%29=x:


g%28x%29=8%2F%28x-10%29 Start with the second function.


g%28f%28x%29%29=8%2F%2810%2B8%2Fx-10%29 Plug in f%28x%29=10%2B8%2Fx


g%28f%28x%29%29=8%2F%288%2Fx%29 Combine like terms.


g%28f%28x%29%29=8%2A%28x%2F8%29 Flip the second lower fraction and multiply.


g%28f%28x%29%29=cross%288%29%2A%28x%2Fcross%288%29%29 Cancel out the common terms.


g%28f%28x%29%29=x Simplify.



Since both f%28g%28x%29%29=x and g%28f%28x%29%29=x, this means that the functions f%28x%29 and g%28x%29 are inverses of one another.