Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=\dfrac{x+3}{x-2}$
$g(x)=\dfrac{2x+3}{x-1}$
a) Verify that the two functions are inverse functions.
Determine $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f\left(\dfrac{2x+3}{x-1}\right)=\dfrac{\dfrac{2x+3}{x-1}+3}{\dfrac{2x+3}{x-1}-2}=\dfrac{\dfrac{2x+3+3x-3}{x-1}}{\dfrac{2x+3-2x+2}{x-1}}=\dfrac{5x}{5}=x$
$(g\circ f)(x)=g(f(x))=g\left(\dfrac{x+3}{x-2}\right)=\dfrac{2\left(\dfrac{x+3}{x-2}\right)+3}{\dfrac{x+3}{x-2}-1}=\dfrac{\dfrac{2x+6+3x-6}{x-2}}{\dfrac{x+3-x+2}{x-2}}=\dfrac{5x}{5}=x$
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.
