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

Answer 1

You need to take the whole function of g(x) and plug it into every single x within f(x). Bit tricky, so it just means we need to be careful. Alright, so this is the first set-up−2x−7x−1−7−2x−7x−1+2So what You should do is make the top portion and the bottom portion all into one fraction. This means I need a common denominator. So to do this, I'll take x-1 as a common denominator and multiply it up into the numerator and the imaginary 1 denominator of -7 and 2. Doing this I have:−2x−7−7(x−1)x−1−2x−7+2(x−1)x−1From here if I divide the top fraction and the bottom fraction, (x-1) will cancel out dueto me flipping the bottom fraction and multiplying. So that leaves me with:−2x−7−7(x−1)−2x−7+2(x−1)Now if I multiply everything out and combine like terms I will have:−2x−7−7x+7−2x−7+2x−2=−9x−9=xSo that is the first part. We have to check both ways to confirm they are actually inverses of each other, though. Take a look at this part and see if you can make sense of what I did.