f(x)=\(\frac{x-7}{x+3}\)
g(x)=\(\frac{-3x-7}{x-1}\)
1. f(g(x))=x
f(g(x)) means to put g(x) wherever you find x in f(x).
f(g(x)) = \(\frac{ \frac{-3x-7}{x-1}-7 }{ \frac{-3x-7}{x-1} +3}\)
f(g(x)) = \(\frac{ \frac{-3x-7-7(x-1)}{x-1} }{ \frac{-3x-7+3(x-1)}{x-1} }\)
Reverse the dominator and simplify with x-1
f(g(x)) = \(\frac{-3x-7-7x+7}{-3x-7+3x-3}\)
f(g(x)) = \(\frac{-10x}{-10}\) = x
So we proved that f(g(x))=x
2. g(f(x))=x
g(f(x)) means to put f(x) wherever you find x in g(x).
g(f(x)) = \(\frac{-3( \frac{x-7}{x+3})-7}{ \frac{x-7}{x+3}-1 }\)
g(f(x)) = \(\frac{ \frac{-3x+21}{x+3} -7}{ \frac{x-7-x-3}{x+3} }\)
g(f(x)) = \(\frac{ \frac{-3x+21-7x-21}{x+3} }{ \frac{x-7-x-3}{x+3} }\)
Reverse the dominator and simplify with x+3
g(f(x))= \(\frac{-10x}{-10} =x\)
So we proved that g(f(x))=x
Because f(g(x)) = x and g(f(x)) = x f and g are inverse.
