Answer:
[tex]\displaystyle f^{-1}(x)=\frac{3+x}{x-2}[/tex]
Step-by-step explanation:
We have the function:
[tex]\displaystyle f(x)=\frac{2x+3}{x-1}[/tex]
Find the inverse of f(x). First, we call y=f(x):
[tex]\displaystyle y=\frac{2x+3}{x-1}[/tex]
We have to solve for x. Multiply by x-1:
[tex]y(x-1)=2x+3[/tex]
Operate:
[tex]yx-y=2x+3[/tex]
Join all the x's to the left side and move the rest to the right side:
[tex]yx-2x=3+y[/tex]
Factor:
[tex]x(y-2)=3+y[/tex]
Solve for x:
[tex]\displaystyle x=\frac{3+y}{y-2}[/tex]
Interchange the variables:
[tex]\displaystyle y=\frac{3+x}{x-2}[/tex]
This is the inverse function:
[tex]\boxed{\displaystyle f^{-1}(x)=\frac{3+x}{x-2}}[/tex]
