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