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. Step 1: To find the inverse of the function \( f(x) = 2x \), start by replacing \( f(x) \) with \( y \). So, we have \( y = 2x \).
2. Step 2: Swap \( x \) and \( y \) to find the inverse. This gives us \( x = 2y \).
3. Step 3: Solve for \( y \) in terms of \( x \). Divide both sides by 2 to get \( y = \frac{x}{2} \).
4. Step 4: Replace \( y \) with \( f^{-1}(x) \) to express the inverse function. Thus, \( f^{-1}(x) = \frac{x}{2} \).
5. Step 5: Verify the inverse by checking \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(x)) \), substitute \( f^{-1}(x) \) into \( f(x) \) to get \( f(\frac{x}{2}) = 2(\frac{x}{2}) = x \). For \( f^{-1}(f(x)) \), substitute \( f(x) \) into \( f^{-1}(x) \) to get \( f^{-1}(2x) = \frac{2x}{2} = x \).

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output is produced by exactly one input. This property ensures that the function has an inverse, as it allows us to uniquely map each value in the range back to a value in the domain. For example, the function f(x) = 2x is one-to-one because no two different x-values will yield the same f(x) value.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y and returns the original input x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y, ensuring that the function is one-to-one to guarantee the existence of the inverse.

Verification of Inverse Functions

To verify that two functions are inverses of each other, we check two conditions: f(f^-1(x)) = x and f^-1(f(x)) = x. This means that applying the original function to its inverse returns the input value, and vice versa. This verification is crucial as it confirms that the derived inverse function accurately undoes the original function's operation.