Problem 37 - A Graphical Approach to College Algebra

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=3 x-7, \quad g(x)=\frac{x+7}{3}$$

Step-by-Step Solution

Answer

\(f(x)\) and \(g(x)\) are inverses since \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Recall the Definition of Inverse Functions

For two functions, \(f\) and \(g\), to be inverses, \(f(g(x)) = x\) and \(g(f(x)) = x\) must hold true for all \(x\). This means if you apply \(g\) to \(f(x)\), you should get \(x\) back, and similarly for applying \(f\) to \(g(x)\).

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

Substitute \(g(x) = \frac{x+7}{3}\) into \(f\). That is, calculate \(f(g(x)) = 3\left(\frac{x+7}{3}\right) - 7\). Simplify this expression:\[ f(g(x)) = 3 \cdot \frac{x+7}{3} - 7 = x + 7 - 7 = x. \]So, \(f(g(x)) = x\).

Step 3: Compute \(g(f(x))\)

Substitute \(f(x) = 3x - 7\) into \(g\). That is, calculate \(g(f(x)) = \frac{3x - 7 + 7}{3}\). Simplify this expression:\[ g(f(x)) = \frac{3x}{3} = x. \]So, \(g(f(x)) = x\).

Step 4: Conclusion

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\) have been shown to be true, we conclude that \(f(x)\) and \(g(x)\) are indeed inverse functions of each other by definition.

Key Concepts

Function Composition

Function composition is a powerful mathematical technique where you apply one function to the results of another. It's like plugging the result of one formula into another formula. In our problem, we have two functions, \( f(x) = 3x - 7 \) and \( g(x) = \frac{x+7}{3} \). To check if they are inverses, we will compose them in two different ways: \( f(g(x)) \) and \( g(f(x)) \).

• First Composition: To find \( f(g(x)) \), we insert \( g(x) = \frac{x+7}{3} \) into \( f(x) \). Simplifying gives us \( 3 \times \frac{x+7}{3} - 7 \), which equals \( x \) after simplifying.
• Second Composition: Now for \( g(f(x)) \), we substitute \( 3x - 7 \) into \( g(x) \). Simplifying \( \frac{3x - 7 + 7}{3} \) also results in \( x \).

Composition allows us to see if applying one function, then the other, eventually returns the original value. This characteristic is key for verifying inverse functions.

Analytical Methods

Analytical methods involve solving problems through a systematic approach of breaking them down into simpler steps. It's like being a detective and solving a mystery by evaluating evidence piece by piece. For our problem, we are proving that two functions are inverses through this clear, logical process.

• Identify and apply the inverse function definition: Begin by understanding that if the functions \( f \) and \( g \) are inverses, composing \( f \) after \( g \) and vice versa should return \( x \).
• Solve step-by-step: For each composition \( f(g(x)) \) and \( g(f(x)) \), perform algebraic steps to verify each equals \( x \). Each step should be simplified carefully.

The advantage of using analytical methods is that they provide a rigorous proof, showing that your conclusion is derived logically and accurately.

Algebraic Proof

An algebraic proof is a chain of logical steps using the rules of algebra to demonstrate mathematical truths. It's a very structured method allowing us to clearly show how ideas connect. In this exercise, our goal is to provide a solid algebraic proof that \( f(x) \) and \( g(x) \) are inverse functions.

Key Steps in the Algebraic Proof:

• Establish Definitions: Start by recalling the definition of inverse functions, where both \( f(g(x)) = x \) and \( g(f(x)) = x \) must hold true for these functions to be inverses.
• Verify Each Composition: Use algebra to simplify \( f(g(x)) \) to \( x \) and \( g(f(x)) \) to \( x \). Each simplification must be executed correctly, checking each operation step-by-step.
• Conclude: If each composition results in \( x \), then by definition, \( f(x) \) and \( g(x) \) are inverses. This conclusion wraps up the algebraic proof neatly.

Algebraic proofs, like this one, are a reliable way to confirm mathematical relationships, ensuring clarity and correctness.