Problem 38 - Algebra 2

Question

Determine whether each pair of functions are inverse functions. \(f(x)=\frac{3 x+2}{7}\) \(g(x)=\frac{7 x-2}{3}\)

Step-by-Step Solution

Answer

The functions \( f(x) \) and \( g(x) \) are inverse functions.

Step 1: Understand the Definition of Inverse Functions

Two functions, \( f(x) \) and \( g(x) \), are inverse functions if \( f(g(x)) = x \) and \( g(f(x)) = x \). This means applying both functions in succession should result in the original input, \( x \).

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

Substitute \( g(x) = \frac{7x - 2}{3} \) into the function \( f(x) \): \[ f(g(x)) = f\left(\frac{7x - 2}{3}\right) = \frac{3\left(\frac{7x - 2}{3}\right) + 2}{7}\]Simplify this expression:First, distribute the 3 inside the numerator: \[ = \frac{3 \times \frac{7x - 2}{3} + 2}{7} = \frac{7x - 2 + 2}{7} \]That simplifies to: \[ = \frac{7x}{7} = x \]

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

Substitute \( f(x) = \frac{3x + 2}{7} \) into the function \( g(x) \):\[ g(f(x)) = g\left(\frac{3x + 2}{7}\right) = \frac{7\left(\frac{3x + 2}{7}\right) - 2}{3} \]Simplify this expression:First, distribute the 7 inside the numerator:\[ = \frac{7 \times \frac{3x + 2}{7} - 2}{3} = \frac{3x + 2 - 2}{3} \]That simplifies to:\[ = \frac{3x}{3} = x \]

Step 4: Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are inverses of each other.

Key Concepts

Function Composition

Function composition is a process of combining two functions, which involves applying one function to the results of another. To understand composition, think of a function as a machine that transforms an input into an output. For example, if you have the functions \( f(x) \) and \( g(x) \), composing \( f \) with \( g \) involves inserting the output of \( g(x) \) into \( f \). Here, the composition is represented as \( f(g(x)) \).In mathematical notation:

• First apply function \( g \) to \( x \), yielding \( g(x) \).
• Then apply function \( f \) to \( g(x) \).
• The result is \( f(g(x)) \).

This process is essential for checking if two functions are inverses. For inverse functions, performing both compositions \( f(g(x)) \) and \( g(f(x)) \) should simply return the original input \( x \). This property helps verify if two functions are indeed inverses.

Algebraic Simplification

Algebraic simplification involves breaking down complex expressions into simpler forms. This allows for easier manipulation and calculation with functions. Let's take a closer look at how simplification works using function composition.For example, consider \( f(g(x)) = \frac{3(\frac{7x - 2}{3}) + 2}{7} \). To simplify:

• First, distribute the multiplication across the addition inside the brackets: \( 3 \times (\frac{7x - 2}{3}) \).
• The fraction simplifies because the 3 in the numerator and denominator cancel out, giving \( 7x - 2 \).
• Add 2 to \( -2 \), resulting in \( 7x \).
• Finally, divide by 7 to result in \( x \), as all terms simplify to the original input \( x \).

This simplification confirms part of the inverse relationship. By simplifying algebraic expressions step by step, you can clearly understand how each component of the expression contributes to the overall function.

Function Properties

Function properties are characteristics or rules that functions follow. Understanding these properties can greatly aid in working with inverse functions. The two essential properties relating to inverse functions are the identity property and the composition property.

• Identity Property: For two functions to be inverses, applying one after the other should return the original value. Mathematically, if \( f \) and \( g \) are inverse functions, then \( f(g(x)) = x \) and \( g(f(x)) = x \).
• Composition Property: As composed earlier, the correct application and simplification must confirm both compositions equal the identity property \( x \).

For inverse functions, these properties guarantee the original input is retained no matter the order of function application. Recognizing these features helps rapidly identify if functions are inverses through computation and conceptual understanding.