Each of Exercises 25–36 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(f^(-1)(x))=f^(-1)(f(x))=x.


f(x) = (x + b) / (x − 2), b > −2 and constant

Verified step by step guidance

1. Start by writing the function explicitly: \(f(x) = \frac{x + b}{x - 2}\), where \(b > -2\) is a constant.
2. To find the inverse function \(f^{-1}(x)\), set \(y = \frac{x + b}{x - 2}\) and then interchange \(x\) and \(y\) to get \(x = \frac{y + b}{y - 2}\).
3. Solve the equation \(x = \frac{y + b}{y - 2}\) for \(y\): multiply both sides by \((y - 2)\) to get \(x(y - 2) = y + b\), then expand and rearrange terms to isolate \(y\).
4. Express \(y\) in terms of \(x\) to find \(f^{-1}(x)\) explicitly. This will give you the formula for the inverse function.
5. Determine the domain and range of \(f^{-1}\) by considering the range and domain of \(f\), respectively, and verify the inverse relationship by checking that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y.

Domain and Range of Functions and Their Inverses

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For inverse functions, the domain and range swap roles: the domain of f becomes the range of f⁻¹, and vice versa. Identifying these sets ensures the inverse is well-defined.

Rational Functions and Restrictions

A rational function is a ratio of polynomials, which may have restrictions where the denominator is zero. For f(x) = (x + b)/(x − 2), x ≠ 2 to avoid division by zero. Such restrictions affect the domain and range and must be considered when finding inverses.