Problem 19 - Precalculus with Limits

Question

In Exercises 19-22, verify that \(f\) and \(g\) are inverse functions. \(f(x) = -\frac{7}{2}x - 3\), \(g(x) = -\frac{2x+6}{7}\)

Step-by-Step Solution

Answer

Yes, \(f\) and \(g\) are inverse functions.

Step 1: Compute g(f(x))

Substitute \(f(x)\) into \(g(x)\) to get \(g(f(x))\). The function \(f(x)\) is given as \(-\frac{7}{2}x - 3\). So, we compute \(g(f(x)) = -\frac{2(-\frac{7}{2}x - 3)+6}{7}\). Simplifying this expression gives \(x\).

Step 2: Compute f(g(x))

Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\). The function \(g(x)\) is given as \(-\frac{2x+6}{7}\). So, we compute \(f(g(x)) = -\frac{7}{2}(-\frac{2x+6}{7}) - 3\). Simplifying this expression also gives \(x\).

Step 3: Conclusion

Since both \(g(f(x))\) and \(f(g(x))\) came out to be \(x\), it can be concluded that the functions \(f\) and \(g\) are indeed inverses of each other.

Key Concepts

Function Composition

Function composition is a powerful operation in mathematics, where you combine two functions into one. This concept is essential when working with inverse functions, like the ones in our exercise. For instance, if you have two functions, say \( f(x) \) and \( g(x) \), composing them results in a new function where you apply one function to the result of the other.

• The composition \( g(f(x)) \) means that you take \( f(x) \) and use its result as an input for \( g(x) \).
• Similarly, \( f(g(x)) \) means using the result of \( g(x) \) as an input for \( f(x) \).

To verify if two functions are inverses, we should get back the original input, \( x \), after composing them in both orders: \( g(f(x)) = x \) and \( f(g(x)) = x \). This shows the functions undo each other’s effect.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions. It's a critical skill, especially when verifying inverse functions, which requires detailed computations. In our case, we manipulated the functions \( f(x) = -\frac{7}{2}x - 3 \) and \( g(x) = -\frac{2x+6}{7} \). Here’s how it worked:

• For \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \), getting \(-\frac{2(-\frac{7}{2}x - 3)+6}{7}\).
• This simplifies step-by-step. Distribute, combine like terms, and simplify fractions until you reach \( x \).

Similarly, for \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \). Manipulate the algebra to simplify the expression back to \( x \). Every algebra step should affirm the logical flow, ensuring errors are avoided as mistakes can lead to incorrect conclusions.

Function Verification

Function verification is like a mathematical detective work where you're checking if your conclusions are correct. When dealing with inverse functions, it's important to verify they truly cancel each other’s effects. The method involves:

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

If after composing both functions in either order you derive the same result, \( x \), then the functions are indeed inverses of each other. This step isn’t just a formality. It confirms the precision of your algebraic manipulations and compositions. Finally, always conclude your process with confidence that both conditions met strongly affirm the functions are inverses, validating your calculations. It's this double-checking process that makes inverse function verification a reliable and necessary procedure in mathematics.