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) = 2x

Verified step by step guidance

1. Start with the given function: \(f(x) = 2x\). To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\), so you have \(y = 2x\).
2. Next, interchange the variables \(x\) and \(y\) to reflect the inverse relationship. This gives \(x = 2y\).
3. Now, solve this equation for \(y\) to express the inverse function. Divide both sides by 2 to isolate \(y\): \(y = \frac{x}{2}\).
4. Rewrite \(y\) as \(f^{-1}(x)\) to get the inverse function: \(f^{-1}(x) = \frac{x}{2}\).
5. To verify the inverse, compute \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). Substitute \(f^{-1}(x)\) into \(f(x)\) and simplify, then substitute \(f(x)\) into \(f^{-1}(x)\) and simplify. Both should simplify to \(x\) if the inverse is correct.

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, effectively 'undoing' the original function. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, meaning applying one after the other returns the original value.

Verification of Inverse Functions

To verify that two functions are inverses, you must show that composing them in both orders returns the input variable. Specifically, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x must hold true for all x in the domains of the respective functions.