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) = (x+2)³

Verified step by step guidance

1. Start with the given function: \(f(x) = (x + 2)^3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = (x + 2)^3\).
2. Swap the variables \(x\) and \(y\) to begin finding the inverse: \(x = (y + 2)^3\).
3. Solve the equation for \(y\) by taking the cube root of both sides: \(\sqrt[3]{x} = y + 2\).
4. Isolate \(y\) by subtracting 2 from both sides: \(y = \sqrt[3]{x} - 2\). This expression represents the inverse function, so write \(f^{-1}(x) = \sqrt[3]{x} - 2\).
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, ensuring no two different inputs share the same output. This property is essential for a function to have an inverse, as the inverse must reverse the mapping uniquely.

Inverse Functions

The inverse of a function f, denoted f⁻¹, reverses the effect of f, swapping inputs and outputs. To find f⁻¹(x), solve the equation y = f(x) for x in terms of y, then interchange variables. The inverse exists only if f is one-to-one.

Verification of Inverse Functions

To verify that two functions are inverses, show that composing them returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms that each function undoes the action of the other, proving they are true inverses.