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 + 3) / (x − 2)

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Start by writing the function as an equation with y: \(y = \frac{x + 3}{x - 2}\).

To find the inverse function \(f^{-1}(x)\), swap the roles of \(x\) and \(y\): \(x = \frac{y + 3}{y - 2}\).

Solve this equation for \(y\) in terms of \(x\): multiply both sides by \((y - 2)\) to get \(x(y - 2) = y + 3\), then expand and rearrange terms to isolate \(y\).

Express \(y\) explicitly as a function of \(x\) to obtain \(f^{-1}(x)\).

Determine the domain and range of \(f^{-1}\) by considering the domain and range of the original function \(f\), and verify the inverse by checking that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

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주요 개념

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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 the equation 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.

Verification of Inverse Functions

To confirm two functions are inverses, compose them in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to the identity function x. This step verifies the correctness of the inverse function and ensures no algebraic errors occurred.

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 + 3) / (x − 2)