Question

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

f(x) = (x-7)/(x+3)
g(x) = (-3x-7)/(x-1)
So:
g(f(x)) = -3(x-7/x+3)-7/(x-7/x+3)-1
f(g(x)) = (-3x-7/x-1)-7/(-3x-7/x-1)+3

But what now? I know it's something obvious, I just can't find a way to simplify.

Answer 1

simplify

Answer 2

How? The only thing I can find is:

g(f(x)) = (-3x+21)/x+3)-7/(x-7/x+3)-1

Answer 3

-3(x-7/x+3)-7/(x-7/x+3)-1
\[\frac{-3\frac{x-7}{x+3}-7}{\frac{x-7}{x+3}-1}=\]
\[\frac{\frac{-3x+21-7x-21}{x+3}}{\frac{x-7-x-3}{x+3}}=\]
\[\frac{-10x}{-10}=x\]

Answer 4

@flixoe make sense?

Answer 5

Yes that makes sense. I didn't think of creating common denominators within both the numerator and denominator. Thank you!

Answer 6

yeah and then
\[\frac{\frac{a}{b}}{\frac{c}{b}}=\frac{a}{c}\]