Find the inverse of f(x) = x^3 + 2

Verified step by step guidance

1. Step 1: Set the function equal to y, so y = f(x) = x^3 + 2.
2. Step 2: To find the inverse, solve for x in terms of y. Start by isolating the x term: y - 2 = x^3.
3. Step 3: Take the cube root of both sides to solve for x, resulting in x = \sqrt[3]{y - 2}.
4. Step 4: Replace y with x in the equation to express the inverse function: f^{-1}(x) = \sqrt[3]{x - 2}.
5. Step 5: Verify the inverse by checking if f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(x), its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. Understanding how to find an inverse function is crucial for solving problems that require reversing operations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to isolate variables. In finding the inverse of a function, one typically swaps the roles of x and y, then solves for y. Mastery of algebraic techniques, such as factoring, expanding, and using properties of equality, is essential for successfully deriving the inverse.

Cubic Functions

Cubic functions are polynomial functions of degree three, characterized by their general form f(x) = ax³ + bx² + cx + d. They can exhibit unique properties such as having one or three real roots. Understanding the behavior of cubic functions, including their increasing and decreasing intervals, is important when determining their inverses, as it affects the function's one-to-one nature.