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) = 1/x

Verified step by step guidance

1. Step 1: To find the inverse of the function \( f(x) = \frac{1}{x} \), start by replacing \( f(x) \) with \( y \). So, \( y = \frac{1}{x} \).
2. Step 2: Swap \( x \) and \( y \) to find the inverse. This gives \( x = \frac{1}{y} \).
3. Step 3: Solve for \( y \) in terms of \( x \). Multiply both sides by \( y \) to get \( xy = 1 \).
4. Step 4: Divide both sides by \( x \) to isolate \( y \). This results in \( y = \frac{1}{x} \). Therefore, \( f^{-1}(x) = \frac{1}{x} \).
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 \( f(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x \). Similarly, for \( f^{-1}(f(x)) \), substitute \( f(x) \) into \( f^{-1}(x) \) to get \( f^{-1}(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x \). Both verifications confirm the inverse is correct.

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 means that if f(a) = f(b), then a must equal b. One-to-one functions have unique inverses, which is crucial for finding the inverse function f^-1(x). Understanding this property ensures that we can correctly derive and verify the inverse.

Inverse Functions

An inverse function reverses the effect of the original function. For a function f(x), its inverse f^-1(x) satisfies the equations f(f^-1(x)) = x and f^-1(f(x)) = x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y. This concept is essential for part a of the question, where we need to derive f^-1(x).

Verification of Inverses

Verifying that two functions are inverses involves showing that applying one function to the result of the other returns the original input. This is done through the equations f(f^-1(x)) = x and f^-1(f(x)) = x. This verification is crucial for part b of the question, as it confirms the correctness of the derived inverse function.