Find the inverse of f(x)= 2x+3/x-1

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]

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