Answer:
[tex]\displaystyle g^{-1}(x)=\frac{-7+2x}{x-2}[/tex]
Step-by-step explanation:
We are given the function:
[tex]\displaystyle g(x)=-\frac{3}{x-2}+2[/tex]
Let's find the inverse of g.
Call y=g(x):
[tex]\displaystyle y=-\frac{3}{x-2}+2[/tex]
We need to solve for x. Multiply both sides by x-2 to eliminate denominators:
[tex]y(x-2)=-3+2(x-2)[/tex]
Operate:
[tex]yx-2y=-3+2x-4[/tex]
Collect the x's to the left side and the rest to the right side of the equation:
[tex]yx-2x=-3-4+2y[/tex]
Factor the left side and operate on the right side:
[tex]x(y-2)=-7+2y[/tex]
Solve for x:
[tex]\displaystyle x=\frac{-7+2y}{y-2}[/tex]
Interchange variables:
[tex]\displaystyle y=\frac{-7+2x}{x-2}[/tex]
Call y as the inverse function:
[tex]\boxed{\displaystyle g^{-1}(x)=\frac{-7+2x}{x-2}}[/tex]
