Question

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

f(x) = (x-9)/(x+5) and g(x) = (-5x-9)/(x-1)

Answer 1

\[f(x)=\frac{x-9}{x+5}\\
g(x)=-\frac{5x+9}{x-1}\]
\[\begin{align*}f(g(x))&=\frac{\left(-\dfrac{5x+9}{x-1}\right)-9}{\left(-\dfrac{5x+9}{x-1}\right)+5}\\\\\\
&=-\frac{\dfrac{5x+9}{x-1}+9}{5-\dfrac{5x+9}{x-1}}\\\\\\
&=-\frac{x-1}{x-1}\cdot\frac{(5x+9)+9(x-1)}{5(x-1)-(5x+9)}\\\\\\
&=-\frac{14x}{-14}\\\\\\
&=x\end{align*}\]
Showing \(g(f(x))=x\) is a very similar process.