Problem 10 - Precalculus : Building Concepts and Connections

Question

Verify that the given functions are inverses of each other. $$f(x)=x+7 ; g(x)=x-7$$

Step-by-Step Solution

Answer

Yes, the functions \( f(x) = x + 7 \) and \( g(x) = x - 7 \) are inverses of each other.

Step 1: Verify \( f(g(x)) = x \)

Keep \( f(x) = x + 7 \) and replace \( x \) with \( g(x) \). The resulting function will be \( f(g(x)) = g(x) + 7 \). Now, replace \( g(x) \) with \( x - 7 \) (since \( g(x) = x - 7 \) ). Then, \( f(g(x)) = (x - 7) + 7 \). By simplifying, \( f(g(x)) = x \). This means that the function \( g(x) = x - 7 \) is indeed the inverse of \( f(x) \).

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

Keep \( g(x) = x - 7 \) and replace \( x \) with \( f(x) \). The resulting function will be \( g(f(x)) = f(x) - 7 \). Now, replace \( f(x) \) with \( x + 7 \) (since \( f(x) = x + 7 \) ). Then, \( g(f(x)) = (x + 7) - 7 \). Simplifying the equation yields: \( g(f(x)) = x \). Hence, the function \( f(x) = x + 7 \) is indeed the inverse of \( g(x) \).

Step 3: Conclusion

After Steps 1 and 2, it has been verified that both \( f(g(x)) = x \) and \( g(f(x)) = x \). So, by definition, the functions \( f(x) = x + 7 \) and \( g(x) = x - 7 \) are inverses of each other.

Key Concepts

Function Composition

Function composition is a useful process where you apply one function to the results of another function. Here, we have two functions: \( f(x) = x + 7 \) and \( g(x) = x - 7 \). To verify they are inverses, you perform function composition in both directions: \( f(g(x)) \) and \( g(f(x)) \).

• In the composition \( f(g(x)) \), you substitute \( g(x) = x - 7 \) into \( f(x) \), giving \( f(g(x)) = (x - 7) + 7 \).
• For \( g(f(x)) \), substitute \( f(x) = x + 7 \) into \( g(x) \), yielding \( g(f(x)) = (x + 7) - 7 \).

The goal is to simplify the expressions to see if they equal \( x \). When both yield \( x \), the functions are confirmed to be inverses of each other. This shows us the power of function composition as a verification tool.

Mathematical Verification

Mathematical verification involves proving or checking the validity of a mathematical statement. For inverse functions, it means showing that two functions undo each other through function composition. If both compositions return the original input, the functions are inverses.
To verify \( f(g(x)) = x \), substitute \( x - 7 \) into \( f(x) \).

• Result: \( f(g(x)) = (x - 7) + 7 \).
• Simplification: \( f(g(x)) = x \).

To verify \( g(f(x)) = x \), substitute \( x + 7 \) into \( g(x) \).

• Result: \( g(f(x)) = (x + 7) - 7 \).
• Simplification: \( g(f(x)) = x \).

This dual verification process assures us that each function truly undoes the other, proving their inverse relationship.

Algebra

Algebra is crucial for simplifying and manipulating algebraic expressions during the verification of inverse functions. Understanding basic algebraic principles allows us to step through the function composition process smoothly.
By applying algebraic operations:

• Addition and subtraction in \( f(g(x)) = (x - 7) + 7 \) simplify to \( x \).
• Similarly, adding and subtracting in \( g(f(x)) = (x + 7) - 7 \) also simplify to \( x \).

These operations rely on the properties of numbers and highlight how inverses function. Each operation cancels the effect of the other, demonstrating how algebra ensures accuracy and validity in mathematical tasks.