Problem 21 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=x-6, \quad g(x)=x+6\)

Step-by-Step Solution

Answer

Yes, \( f(x) = x - 6 \) and \( g(x) = x + 6 \) are inverses of each other.

Step 1: Understand the Inverse Function Property

Two functions, \( f(x) \) and \( g(x) \), are inverses of each other if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domain of \( g \) and \( f \) respectively. We need to verify both these conditions.

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

Substitute \( g(x) = x+6 \) into \( f(x) = x-6 \). Thus, calculate \( f(g(x)) = f(x+6) = (x+6) - 6 = x \). This shows that \( f(g(x)) = x \).

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

Substitute \( f(x) = x-6 \) into \( g(x) = x+6 \). Thus, calculate \( g(f(x)) = g(x-6) = (x-6) + 6 = x \). This shows that \( g(f(x)) = x \).

Step 4: Conclusion

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, \( f(x) = x - 6 \) and \( g(x) = x + 6 \) are indeed inverses of each other.

Key Concepts

Inverse Function Property

The Inverse Function Property is a fundamental concept in mathematics that helps us determine if two functions are inverses of each other. To verify this, we need to see if applying one function to the result of the other function returns the original input. In other words, we need to check the following conditions:

• Calculate \( f(g(x)) \): If this equals \( x \), then part of the condition is satisfied.
• Calculate \( g(f(x)) \): If this also equals \( x \), then the functions are inverses.

Both conditions must hold true to confirm that \( f(x) \) and \( g(x) \) are inverses. In our problem, we demonstrated that \( f(g(x)) = x \) and \( g(f(x)) = x \), so \( f(x) = x - 6 \) and \( g(x) = x + 6 \) are inverses of each other. This property assures us that applying \( f \) to \( g \) and vice versa yields the input we started with.

Function Composition

Function composition involves applying one function to the result of another function. This is symbolized by \( f(g(x)) \) or \( g(f(x)) \). It's an essential concept in understanding inverse functions because it allows us to check if one function "undoes" the action of another. If \( f \) and \( g \) undo each other, they are inverses, meaning their composition results recycle back to the original input.For example, in the exercise:

• By composing \( f(g(x)) \), we get \( f(x+6) = x \).
• Similarly, composing \( g(f(x)) \) gives us \( g(x-6) = x \).

These equalities confirm that each function effectively reverses the effect of the other. Function composition is an instrumental tool in working out complex algebraic functions and plays a critical role in nearly all areas of mathematics.

Algebraic Functions

Algebraic functions are defined by combining constant and variable expressions using arithmetic operations. These functions can take many forms, including linear, quadratic, polynomial, and more. In the exercise, we dealt with linear functions:

• \( f(x) = x - 6 \)
• \( g(x) = x + 6 \)

These functions are straightforward, yet they beautifully illustrate key principles of inverse functions.
To solve for algebraic inverses, simply perform operations that mathematically reverse each other. In our exercise:

• \( f(x) = x - 6 \) moves every input \( x \) by subtracting 6.
• \( g(x) = x + 6 \) compensates by adding back 6, thus returning \( x \) to its original value.

Understanding algebraic functions is crucial as they form the basis of many mathematical models and are used in real-world problem-solving involving equations.