Answer: [tex]\bold{f^{-1}(x)=\dfrac{2x-1}{2+x}}[/tex]
Step-by-step explanation:
Inverse is when you swap the x's and y's and then manipulate the equation to solve for y.
[tex]y=\dfrac{2x+1}{2-x}\\\\\\\text{Swap the x's and y's:}\\x=\dfrac{2y+1}{2-y}\\\\\\\text{Multiply both sides by 2-y}:\\x(2-y)=2y+1\\\\\\\text{distribute x into 2-y}:\\2x-xy=2y+1\\\\\\\text{subtract 1 from both sides and add xy to both sides}:\\2x-1=2y+xy\\\\\\\text{factor out y from the right side}:\\2x-1=y(2+x)\\\\\\\text{divide 2+x from both sides}:\\\dfrac{2x-1}{2+x}=y\\\\\\\text{Therefore, the inverse }f^{-1}(x)=\dfrac{2x-1}{2+x}[/tex]
