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

Verified step by step guidance

1. Start with the given function: \(f(x) = \sqrt{x}\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = \sqrt{x}\).
2. Next, interchange the variables \(x\) and \(y\) to find the inverse: \(x = \sqrt{y}\). This step reflects the idea that the inverse function swaps the roles of inputs and outputs.
3. Now, solve the equation \(x = \sqrt{y}\) for \(y\). Square both sides to eliminate the square root: \(x^2 = y\).
4. Rewrite \(y\) as \(f^{-1}(x)\) to express the inverse function explicitly: \(f^{-1}(x) = x^2\). Remember to consider the domain restrictions since the original function outputs only non-negative values.
5. To verify the inverse, compute the compositions \(f(f^{-1}(x))\) 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 (injective) 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 because it guarantees that the inverse will 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 equation y = f(x) for x in terms of y and then swapping variables.

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 that the functions undo each other's operations perfectly.