Find the inverse of f(x) = x³ + 2

Verified step by step guidance

1. Start with the function given: \(f(x) = x^3 + 2\). To find the inverse, replace \(f(x)\) with \(y\), so we have \(y = x^3 + 2\).
2. Swap the variables \(x\) and \(y\) to begin finding the inverse function. This gives the equation \(x = y^3 + 2\).
3. Solve the equation \(x = y^3 + 2\) for \(y\). Begin by isolating the cubic term: subtract 2 from both sides to get \(x - 2 = y^3\).
4. Next, take the cube root of both sides to solve for \(y\): \(y = \sqrt[3]{x - 2}\).
5. Finally, express the inverse function using function notation: \(f^{-1}(x) = \sqrt[3]{x - 2}\).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function, meaning if f maps x to y, then its inverse maps y back to x. To find the inverse, you swap the roles of x and y and solve for y. The inverse exists only if the function is one-to-one.

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input, ensuring the function passes the horizontal line test. This property is essential for the existence of an inverse function, as it guarantees that the inverse will also be a function.

Solving Equations Involving Cubes

To find the inverse of f(x) = x^3 + 2, you need to solve for x in terms of y by isolating the cubic term and then taking the cube root. Understanding how to manipulate and solve cubic equations is crucial for expressing the inverse function explicitly.