Answer:
[tex]f^{-1}(x)=\frac{2x+7}{x-3}[/tex].
Step-by-step explanation:
The given function is [tex]f(x)=\frac{3x+7}{x-2},\:x\ne2[/tex].
We want to find a rule for [tex]f^{-1}(x)[/tex].
Let [tex]y=\frac{3x+7}{x-2}[/tex].
Interchange x and y.
[tex]x=\frac{3y+7}{y-2}[/tex].
Solve for y.
[tex]x(y-2)=3y+7[/tex].
Expand
[tex]xy-2x=3y+7[/tex].
Group y-terms
[tex]xy-3y=7+2x[/tex].
Factor
[tex](x-3)y=7+2x[/tex].
[tex]y=\frac{2x+7}{x-3}[/tex].
[tex]\therefore f^{-1}(x)=\frac{2x+7}{x-3},\:x\ne3[/tex].
