Problem 20 - Algebra for College Students

Question

Verify that the two given functions are inverses of each other. $$ f(x)=-3 x+4 \text { and } g(x)=\frac{4-x}{3} $$

Step-by-Step Solution

Answer

The functions are inverses of each other because both \( f(g(x)) = x \) and \( g(f(x)) = x \).

Step 1: Introduction

To verify if two functions, \( f(x) = -3x + 4 \) and \( g(x) = \frac{4-x}{3} \), are inverses of each other, we need to check two conditions:1. \( f(g(x)) = x \)2. \( g(f(x)) = x \)

Step 1: Calculate \( f(g(x)) \)

First, substitute \( g(x) \) into \( f(x) \):\[ f(g(x)) = f\left(\frac{4-x}{3}\right) = -3\left(\frac{4-x}{3}\right) + 4 \]Simplify the expression:\[ = -3 \cdot \frac{4-x}{3} + 4 = -(4-x) + 4 \]\[ = -4 + x + 4 \]\[ = x \]Thus, \( f(g(x)) = x \).

Step 2: Calculate \( g(f(x)) \)

Next, substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(-3x + 4) = \frac{4 - (-3x + 4)}{3} \]Simplify the expression:\[ = \frac{4 + 3x - 4}{3} = \frac{3x}{3} \]\[ = x \]Thus, \( g(f(x)) = x \).

Step 4: Conclusion

Since \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) = -3x + 4 \) and \( g(x) = \frac{4-x}{3} \) are indeed inverses of each other.

Key Concepts

Function Composition

Understanding function composition is essential when verifying inverse functions. When we combine two functions, say \( f(x) \) and \( g(x) \), through composition, we substitute one into the other. For example, the expression \( f(g(x)) \) means we take the output of \( g(x) \) and use it as the input for \( f(x) \). This process helps us in checking the relationships between functions.

• To verify if two functions are inverses, we perform two compositions: \( f(g(x)) \) and \( g(f(x)) \).
• If both result in the original input \( x \), then the functions are inverses of each other.

Function composition is not only useful for inverses but also in other areas of mathematics, making it a fundamental concept.

Algebraic Verification

When verifying inverse functions algebraically, we rely on substituting and simplifying expressions. The key objective is to prove that the composite functions \( f(g(x)) \) and \( g(f(x)) \) both simplify to \( x \).
For instance, given \( f(x) = -3x + 4 \) and \( g(x) = \frac{4-x}{3} \), when substituting \( g(x) \) into \( f(x) \), we have:

• \( f(g(x)) = f\left(\frac{4-x}{3}\right) = -3\left(\frac{4-x}{3}\right) + 4 \) leads to \( x \) after simplification.
• Similarly, \( g(f(x)) = \frac{4 - (-3x + 4)}{3} \) also results in \( x \).

These steps verify algebraically that the functions are inverses. Careful simplification and substitution are crucial for accurate results.

Inverse Properties

Inverse functions have special properties that define their unique relationship. The inverse of a function \( f(x) \) typically denoted as \( f^{-1}(x) \), performs the opposite operation. It's like undoing what \( f(x) \) does.

• The identity that \( f(g(x)) = x \) and \( g(f(x)) = x \) is central to inverses.
• Functionally, applying \( f(x) \) and \( f^{-1}(x) \) consecutively brings us back to the starting point \( x \).

This property is immensely useful not just in pure mathematics but in real-world applications where reversing a process step-by-step is required. Knowing how to identify and verify these properties strengthens problem-solving skills in various mathematical contexts.