Problem 22 - 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

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

Step 1: Understand the Inverse Function Property

To show that two functions are inverses of each other, we need to verify two conditions: \(f(g(x)) = x\) and \(g(f(x)) = x\). This means that applying one function to the result of another should bring us back to the original value \(x\).

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

First, substitute \(g(x) = \frac{x}{3}\) into \(f(x) = 3x\). Calculate \(f(g(x)):\)\[f(g(x)) = f\left(\frac{x}{3}\right) = 3 \cdot \frac{x}{3}.\]Simplifying this gives:\[f(g(x)) = x.\]

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

Next, substitute \(f(x) = 3x\) into \(g(x) = \frac{x}{3}\). Calculate \(g(f(x)):\)\[g(f(x)) = g(3x) = \frac{3x}{3}.\]Simplifying this gives:\[g(f(x)) = x.\]

Step 4: Conclusion

Since both conditions \(f(g(x)) = x\) and \(g(f(x)) = x\) are satisfied, the functions \(f\) and \(g\) are inverses of each other by the definition of inverse functions.

Key Concepts

Inverse Function Property

The Inverse Function Property is an essential concept that helps us understand if two functions are inverses of one another. In order for two functions, say \( f \) and \( g \), to be inverses, we need to check two critical conditions:

• \( f(g(x)) = x \)
• \( g(f(x)) = x \)

These conditions basically tell us that if you apply one function and then the other, you should always get back to your starting number \( x \). So, if either of these conditions doesn't hold true, then \( f \) and \( g \) aren't actually inverses of each other.
In the case of our exercise, we looked at \( f(x) = 3x \) and \( g(x) = \frac{x}{3} \). By plugging one into the other in the manner described, and confirming that both conditions give us back \( x \), we proved they are indeed inverses.

Function Composition

Function Composition is like following a recipe: first, you start with one operation, and then you perform another one, step by step. It’s the process of combining two functions so that the output of one function becomes the input of the other.
In mathematical terms, if you have two functions, say \( f \) and \( g \), their composition is written as \( f(g(x)) \) or \( g(f(x)) \). In simpler terms:

• \( f(g(x)) \) means you apply \( g \) to \( x \), and then \( f \) to the result.
• \( g(f(x)) \) means you apply \( f \) to \( x \), and then \( g \) to the result.

In the exercise, we did this exact operation twice. First, we took \( g(x) = \frac{x}{3} \) and applied \( f \) to it, resulting in \( f(g(x)) = x \). Then we reversed the process by applying \( g \) to \( f(x) = 3x \), which brought us back to \( g(f(x)) = x \). This process of showing each composition equals \( x \) means these two functions cancel each other's effect like perfect undo buttons.

Algebraic Simplification

Algebraic Simplification is the beautiful art of making expressions as straightforward as possible without changing their value. When working with functions and their inverses, simplifying expressions helps in identifying and proving properties more easily.
In our exercise, when we calculated \(f(g(x))\) and \(g(f(x))\), we performed simplification steps:

• For \( f(g(x)) = 3 \cdot \frac{x}{3} \), we simplified this to \( x \) by canceling out the \( 3 \) from the numerator and denominator.
• For \( g(f(x)) = \frac{3x}{3} \), we also simplified to \( x \) using the same cancellation of \( 3 \).

This simplification not only confirms the functions are inverses but also reassures us at every step that we're maintaining equality. It's like cleaning up a puzzle. When each piece fits perfectly, you know you've assembled it correctly! Simplification makes checking and proving conclusions so much clearer and more straightforward.