Expert: Sherry Wallin - 11/15/2009

Question
I only have one problem that I don't understand:

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

f(x)= (x+8)/(x-2) and g(x)= (2x+8)/(x-1)

Answer
Celia~
    I like to think of finding the inverse as a game and part of that game is substituting in another variable for one of the functions so let g(x) = y. Now you want to show that f(g(x)) = f(g(y))
What this means is that you are going to insert g(y) in everywhere you see an x in f(x):
Sometimes it helps to spell out what f(x) says. It says to take your input and add 8 to it and then to divide by the quantity of your input minus 2. So f(g(y)) = [g(y)+8]/[g(y)-2]. You will now need to re-substitute what g(y) really is: [((2x+8)/(x-1))+8]/[((2x+8)/(x-1))-2]. In order to simplify you need to find the LCM which is x-1 and multiply top and bottom by x-1 which gives you:
[(x-1)[(2x+8)/(x-1)]+8(x-1)]/[(x-1)[(2x+8)/(x-1)]-(x-1)2] = [2x+8 +8(x-1)]/[2x+8 -2(x-1)] ->
[2x + 8 + 8x - 8]/[2x + 8 -2x +2]= 10x/10 = x. You will find g(f(x)) in the same way and when you show that g(f(x)) = x you have shown that f and g are inverses of each other.

Math Prof