Problem 26 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=3 x, \quad g(x)=\frac{x}{3} $$

Step-by-Step Solution

Answer

The functions \( f(x) = 3x \) and \( g(x) = \frac{x}{3} \) are inverses of each other.

Step 1: Understand the Inverse Function Property

Two functions \( f \) and \( g \) are inverses if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of \( g \) and \( f \) respectively.

Step 2: Verify \( f(g(x)) = x \)

Find \( f(g(x)) \) by substituting \( g(x) = \frac{x}{3} \) into \( f(x) = 3x \). This gives us: \[f(g(x)) = f\left(\frac{x}{3}\right) = 3 \cdot \frac{x}{3} = x.\]This verifies the first condition of the inverse function property.

Step 3: Verify \( g(f(x)) = x \)

Find \( g(f(x)) \) by substituting \( f(x) = 3x \) into \( g(x) = \frac{x}{3} \). This results in: \[g(f(x)) = g(3x) = \frac{3x}{3} = x.\]This verifies the second condition of the inverse function property.

Step 4: Conclusion

Since both conditions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, the functions \( f(x) = 3x \) and \( g(x) = \frac{x}{3} \) are inverses of each other.

Key Concepts

Function Composition

In mathematics, function composition is a crucial concept to understand the relationship between two functions. When you combine the outputs of one function with the inputs of another, this is known as composing functions. In simpler terms, it involves nesting one function inside another. In our example, we deal with the functions \(f(x) = 3x\) and \(g(x) = \frac{x}{3}\). These two functions need to be composed to test whether they are inverses of each other. When composing these functions, we determine:

• \(f(g(x))\) which is calculated by replacing \(x\) in \(f(x)\) with \(g(x) = \frac{x}{3}\)
• \(g(f(x))\) which involves replacing \(x\) in \(g(x)\) with \(f(x) = 3x\).

This illustrates how outputs from \(g\) become inputs for \(f\) and vice versa, allowing us to verify the inverse relationship.

Inverse Function Property

The inverse function property plays a fundamental role in determining whether two functions are genuinely inverses of each other. Two functions, \(f\) and \(g\), are inverses if they undo each other's processes. This means:

• \(f(g(x)) = x\), where applying \(g\) first and then \(f\) returns the original \(x\)
• \(g(f(x)) = x\), where applying \(f\) first and then \(g\) also returns the original \(x\).

These properties must hold true for all \(x\) within the relevant domains of the functions. If both conditions are satisfied, we can confidently say that the functions are inverses. In our case, \(f(x) = 3x\) and \(g(x) = \frac{x}{3}\) satisfy these properties, confirming their inverse relationship.

Algebraic Verification

Algebraic verification is the process of using algebraic manipulations to confirm certain mathematical properties or results. When examining whether two functions are inverses, this method includes calculating both \(f(g(x))\) and \(g(f(x))\), ensuring each equals \(x\). Here's how it works for our functions: - For \(f(g(x))\), substitute \(g(x) = \frac{x}{3}\) into \(f(x) = 3x\) to get \(f\left(\frac{x}{3}\right) = x\). - For \(g(f(x))\), substitute \(f(x) = 3x\) into \(g(x) = \frac{x}{3}\) to get \(g(3x) = x\).This step-by-step verification ensures that both conditions of the inverse function property are met, thus providing a strong algebraic evidence that \(f\) and \(g\) are inverses. This process is systematic, reduces errors, and ensures clarity in confirming the inverse relationship.