The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = x³ +2

Verified step by step guidance

1. Start with the given function: \(f(x) = x^3 + 2\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x^3 + 2\).
2. Swap the variables \(x\) and \(y\) to begin solving for the inverse: \(x = y^3 + 2\).
3. Isolate the cubic term by subtracting 2 from both sides: \(x - 2 = y^3\).
4. Take the cube root of both sides to solve for \(y\): \(y = \sqrt[3]{x - 2}\). This expression represents the inverse function \(f^{-1}(x)\).
5. To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f(x)\) and simplify to check if it equals \(x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}(x)\) and simplify to check if it equals \(x\).

Key Concepts

One-to-One Functions

A one-to-one function assigns each input a unique output and vice versa, ensuring that no two different inputs produce the same output. This property is essential for a function to have an inverse, as the inverse must also be a function.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, meaning if f(x) maps x to y, then f⁻¹(y) maps y back to x. Finding the inverse involves solving the original function's equation for x in terms of y.

Verification of Inverse Functions

To verify that two functions are inverses, you must show that composing them in either order returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms the functions undo each other's operations.