Problem 21 - Precalculus Essentials

Question

The functions 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=\frac{1}{x}$$

Step-by-Step Solution

Answer

The inverse function for \(f(x) = \frac{1}{x}\) is \(f^{-1}(x) = \frac{1}{x}\). The provided expressions \(f\left(f^{-1}(x)\right)\) and \(f^{-1}(f(x))\) confirm that the inverse function is correct.

Step 1: Finding the Inverse of the Function

To find the inverse of the function \(f(x) = \frac{1}{x}\), replace \(f(x)\) with \(y\). This will give you the equation \(y = \frac{1}{x}\). Next, swap \(x\) and \(y\). This gives us \(x = \frac{1}{y}\). Solving this for \(y\) gives us \(y = \frac{1}{x}\). Thus, the inverse function, \(f^{-1}(x)\), is \(\frac{1}{x}\).

Step 2: Verifying the Inverse Function

To verify the inverse function, ensure that \(f\left(f^{-1}(x)\right) = x\) and \(f^{-1}(f(x)) = x\) hold true. Substitute \(f^{-1}(x)\) into \(f(x)\), which should result in \(x\). For \(f\left(f^{-1}(x)\right)\), you get \(f\left(\frac{1}{x}\right) = x\). Next, substitute \(f(x)\) into \(f^{-1}(x)\), which should also result in \(x\). For \(f^{-1}(f(x))\), you get \(f^{-1}\left(\frac{1}{x}\right) = x\). As each expression reduces to \(x\), the inverse function is verified.

Key Concepts

One-to-One Functions

One-to-one functions are quite special in mathematics. They have a unique property that each input value maps to a unique output value. This means no two different input values will map to the same output. Such functions are always "injective". This uniqueness makes determining their inverses possible.

In simpler terms, if each element of the domain (input) corresponds to only one element of the range (output) and vice versa, then we are dealing with a one-to-one function. A straightforward example is the function \(f(x) = \frac{1}{x}\). You will notice how each input not only affects the function once but results in a unique output.

Understanding one-to-one functions allows one to find inverse functions accurately. By establishing that each element in the domain maps to a unique element in the range, you can readily perform the operation of swapping roles of inputs and outputs, simplifying the process of finding the inverse.

Function Verification

Verifying that a given inverse function is correct is crucial and involves a two-way check: ensuring \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This is simply meant to check that performing one operation after the other brings you back to where you started.

Let's break it down:

• First Check: Substitute the inverse into the original. Apply \(f^{-1}\) to \(x\) and substitute into \(f\). If \(f(f^{-1}(x)) = x\) holds true, this confirms half of the verification.
• Second Check: Reverse the substitution. Apply \(f\) and substitute into \(f^{-1}\). If \(f^{-1}(f(x)) = x\) holds, it confirms the other half.

In the function \(f(x) = \frac{1}{x}\), testing the operations with \(f^{-1}(x) = \frac{1}{x}\) meets both conditions; thus our verification confirms the correctness of the inverse.

Pre-Calculus Problem Solving

When approaching pre-calculus problems, especially involving functions and their inverses, it's essential to have a solid problem-solving strategy. These types of problems often require a clear understanding of algebraic manipulations and function properties.

Here's a step-by-step approach you might find helpful:

• Understand the Problem: Read the problem carefully, identifying what is given and what is needed. Note down the function details and any specific requirements, like finding inverses.
• Find the Inverse: For one-to-one functions, swap the roles of \(x\) and \(y\), then solve for \(y\). This gives you \(f^{-1}(x)\).
• Verify Your Solution: Always verify as earlier; check both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
• Make Sense of the Results: Double-check calculations, ensuring all transformations correctly support the findings.

These strategies ensure accuracy and help mitigate simple errors often encountered in pre-calculus problem solving.