Each of Exercise gives a formula for a function y = ƒ(x)

Chapter 1, Problem 35E

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QUESTION:

Each of Exercise gives a formula for a function \(y=f(x)\). In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1}\). As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\).

\(f(x)=\frac{x+b}{x-2}, \quad b>-2\) and constant

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QUESTION:

Each of Exercise gives a formula for a function \(y=f(x)\). In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1}\). As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\).

\(f(x)=\frac{x+b}{x-2}, \quad b>-2\) and constant

ANSWER:

Step 1 of 4

To determine \(f^{-1}\) put \(y=f(x)\) and solve for \(x\) :

\(\begin{array}{c} y=\frac{x+b}{x-2} \\ (x-2) y=x+b \\ x y-x=2 y+b \\ x=\frac{2 y+b}{y-1} \end{array}\)

Now we will interchange x and y to find the formula for \(f^{-1}\).

\(y=\frac{2 x+b}{x-1}\)

which leads us to conclude that the inverse function \(f^{-1}\) is defined as

\(f^{-1}(x)=\frac{2 x+b}{x-1}\)

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