Problem 25 - 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

Functions \(f(x) = x-6\) and \(g(x) = x+6\) are inverses.

Step 1: Understand the Inverse Function Property

To show that two functions are inverses of each other, we must demonstrate that applying one function to a value and then applying the other returns to us our original value. Mathematically, if \(f\) and \(g\) are inverses, then \(g(f(x)) = x\) and \(f(g(x)) = x\).

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

Start by substituting \(f(x)\) into \(g(x)\). Since \(f(x) = x - 6\), we have:\[g(f(x)) = g(x-6) = (x-6) + 6 = x.\]This confirms that \(g(f(x)) = x\).

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

Next, substitute \(g(x)\) into \(f(x)\). Since \(g(x) = x + 6\), we have:\[f(g(x)) = f(x+6) = (x+6) - 6 = x.\]This confirms that \(f(g(x)) = x\).

Step 4: Conclusion

Since both \(g(f(x)) = x\) and \(f(g(x)) = x\), by the Inverse Function Property, \(f\) and \(g\) are indeed inverse functions of each other.

Key Concepts

Inverse Function Property

The inverse function property is pivotal in understanding how two functions can undo each other's work. When we say that two functions \( f \) and \( g \) are inverses, we mean that they reverse the effects of each other. In mathematical terms, this means that applying one function and then the other should return the original input value. For example, if you have a function \( f(x) \), and you perform another function \( g(x) \) right after, the output should bring you back to your initial \( x \). To confirm this, we need:

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

This automatic reset to the original input is what verifies the inverse nature of two functions. Understanding this is essential when dealing with inverse equations and transformations, as it shows how inputs are looped back to their beginnings.

Composition of Functions

The composition of functions is a technique where we apply two functions in sequence. Visualize this as plugging the output of one function into the input of another. The composition is often denoted by \( g(f(x)) \) or \( f(g(x)) \), which represent different sequences of applying \( f \) and \( g \).

When dealing with potential inverse functions, testing their composition is key:

• Compute \( g(f(x)) \). If \( f(x) = x-6 \), then plugging it into \( g(x) \) results in \( g(x-6) = (x-6)+6 = x \).
• Check \( f(g(x)) \) as well. If \( g(x) = x+6 \), substitute it into \( f(x) \) producing \( f(x+6) = (x+6)-6 = x \).

Both compositions returning to \( x \) help illustrate that \( f \) and \( g \) are inverse functions. Composition effectively acts as a mathematical verification process for relationships between functions.

Proof of Inverse Functions

Providing a proof for inverse functions involves showing that each function undoes the work of the other. In our exercise, the functions \( f(x) = x - 6 \) and \( g(x) = x + 6 \) can easily be verified through this process. Consider these steps as your go-to recipe:

• Start with \( g(f(x)) \): Substituting \( f(x) = x-6 \) into \( g \), you compute \( g(x-6) = x \), proving that the application of \( g \) undoes \( f \).
• Next, validate \( f(g(x)) \): Place \( g(x) \) into \( f \), and you'll see \( f(x+6) = x \). This confirms that \( f \) undoes the alterations made by \( g \).

Both conditions being satisfied, where individual compositions yield \( x \), establishes that these two functions are indeed inverses of each other. This systematic approach can be applied to a wide variety of function pairs, making it a valuable tool in many areas of mathematics.