Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=\dfrac{x-1}{x+5}$
$g(x)=-\dfrac{5x+1}{x-1}$
a) Verify that the two functions are inverse functions.
Determine $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f\left(-\dfrac{5x+1}{x-1}\right)=\dfrac{-\dfrac{5x+1}{x-1}-1}{-\dfrac{5x+1}{x-1}+5}=\dfrac{\dfrac{-5x-1-x+1}{x-1}}{\dfrac{-5x-1+5x-5}{x-1}}=\dfrac{-6x}{-6}=x$
$(g\circ f)(x)=g(f(x))=g\left(\dfrac{x-1}{x+5}\right)=-\dfrac{5\left(\dfrac{x-1}{x+5}\right)+1}{\dfrac{x-1}{x+5}-1}=-\dfrac{\dfrac{5x-5+x+5}{x+5}}{\dfrac{x-1-x-5}{x+5}}=-\dfrac{6x}{-6}=x$
We got:
$(f\circ g)(x)=(g\circ f)(x)=x$,
therefore the two functions are inverse functions.
b) Graph $f$ and $g$ and the line $y=x$.
The graphs of $f$ and $g$ are symmetric with respect to the line $y=x$; therefore $f$ and $g$ are inverse functions.
