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) = x³ + 1

Verified step by step guidance

1. Start by writing the function given: \(f(x) = x^3 + 1\).
2. To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = x^3 + 1\).
3. Swap the roles of \(x\) and \(y\) to find the inverse: \(x = y^3 + 1\).
4. Solve this equation for \(y\) to express \(y\) in terms of \(x\): subtract 1 from both sides to get \(x - 1 = y^3\), then take the cube root to find \(y = \sqrt[3]{x - 1}\).
5. Identify the domain and range of \(f^{-1}(x)\). Since the original function \(f(x) = x^3 + 1\) has domain \((-\infty, \infty)\) and range \((-\infty, \infty)\), the inverse function will have domain \((-\infty, \infty)\) and range \((-\infty, \infty)\) as well.

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 inputs, while the range is the set of all possible outputs. For an inverse function, the domain and range swap roles compared to the original function. Identifying these sets ensures the inverse is properly 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 x, the identity function. This check validates the correctness of the inverse function found.