Question

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

f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quantity negative five x minus nine divided by quantity x minus one.

Answer 1

\(\bf f(x)=\cfrac{x-9}{x+5}\qquad g(x)=\cfrac{-5x-9}{x-1} ?\)

Answer 2

you can always post a screenshot here btw, just by using the [Attach File] blue button, so you know

Answer 3

yes

Answer 4

ok... so if indeed, f(x) and g(x) are inverses of each other
that means
that g(x) is the inverse of f(x)
and that f(x) is therefore the inverse of g(x)
so to verify, you'd do as indicated find f( g(x) ) and that will simplify to "x" if they're indeed inverses of each other

lemme do the first one, so you can do the next one

Answer 5

\(\bf f(x)=\cfrac{x-9}{x+5}\qquad {\color{brown}{ g(x)}}=\cfrac{-5x-9}{x-1}
\\ \quad \\
f(\quad {\color{brown}{ g(x)}}\quad )=\cfrac{{\color{brown}{ g(x)}}-9}{{\color{brown}{ g(x)}}+5}\to
\cfrac{\left({\color{brown}{ \frac{-5x-9}{x-1}}}\right)-9}{\left({\color{brown}{ \frac{-5x-9}{x-1}}}\right)+5}\to
\cfrac{\frac{-5x-9-9(x-1)}{x-1}}{\frac{-5x-9+5(x-1)}{x-1}}
\\ \quad \\
\to \cfrac{\frac{-5x\cancel{ -9 }-9x\cancel{ +9 }}{x-1}}{\frac{\cancel{ -5x }-9\cancel{ +5x }-5}{x-1}}\to
\cfrac{\frac{-14x}{x-1}}{\frac{-14}{x-1}}\to \cfrac{\cancel{ -14 }x}{\cancel{ x-1 }}\cdot \cfrac{\cancel{ x-1 }}{\cancel{ -14 }}\)

notice what's left

so now... try the 2nd one g( f(x) )