Problem 20 - Precalculus with Limits

Question

In Exercises 19-22, verify that \(f\) and \(g\) are inverse functions. \(f(x) = \frac{x-9}{4} - 3\), \(g(x) = 4x + 9\)

Step-by-Step Solution

Answer

The functions \(f(x)\) and \(g(x)\) are not inverse functions.

Step 1: Checking the composition of \(f(g(x))\)

Insert \(g(x)\) into the function expression for \(f(x)\). This becomes: \(f(g(x)) = \frac{g(x)-9}{4} - 3 = \frac{4x+9-9}{4} - 3 = \frac{4x}{4} - 3 = x - 3\)

Step 2: Solving for \(x\) in \(f(g(x))\)

Solve the equation \(x - 3 = x\), gives us \(0 = 3\). This is not true, so the function composition \(f(g(x))\) does not equal \(x\).

Step 3: Checking the composition of \(g(f(x))\)

Insert \(f(x)\) into the function expression for \(g(x)\). This becomes: \(g(f(x)) = 4*f(x) + 9 = 4*\left(\frac{x-9}{4} - 3\right) + 9 = x-9 - 12 + 9 = x - 12\)

Step 4: Solving for \(x\) in \(g(f(x))\)

Solving the equation \(x - 12 = x\), gives us \(0 = 12\). This is not true, so the function composition \(g(f(x))\) does not equal \(x\).

Step 5: Conclusion

Since neither of the function compositions equals \(x\), this indicates that the functions \(f(x)\) and \(g(x)\) are not inverse functions.

Key Concepts

Function Composition

Function composition is a mathematical operation where one function is applied to the result of another function. This process allows us to chain together multiple functions into a single operation. In simpler words, if you have two functions, say \( f \) and \( g \), the composition denoted as \( f(g(x)) \) means you first apply \( g \) to \( x \), and then apply \( f \) to the result of \( g(x) \).

• For instance, if \( g(x) = 4x + 9 \), you substitute \( g(x) \) into \( f(x) \).
• Upon substituting, computation is done to simplify and see if you end up with \( x \) again, which would indicate that the functions are inverses of each other.

Function composition is critical in determining inverse functions. Inverse functions are those that "reverse" each other's operations; that is, when composed together in either order, they should return \( x \).

Algebraic Verification

Algebraic verification involves using algebra to confirm whether two functions are inverses. This is done by checking the composition of the functions to see if they return the identity function. The identity function is essentially \( x \) itself, which means the function does nothing other than return what was input.

• For the given functions \( f(x) = \frac{x-9}{4} - 3 \) and \( g(x) = 4x + 9 \), verification means checking both \( f(g(x))\) and \( g(f(x)) \).
• If neither gives back the original input \( x \), they fail this test, meaning they are not inverses.

This exercise shows that both computations produce expressions \( x - 3 \) and \( x - 12 \), which clearly do not equal \( x \), thus proving algebraically that \( f(x) \) and \( g(x) \) aren't inverse functions.

Precalculus

Precalculus serves as the foundation for concepts used in calculus, including understanding functions, their properties, and how they relate to one another. It strengthens skills in algebra and provides insight into more advanced mathematical theories.

• Inverse functions: Learn how to identify if functions can undo each other’s effects.
• Function composition: Understand how functions interact and operate together.
• Algebra: Develop proficiency in verifying mathematical properties like the identity function using algebraic calculations.

In precalculus, grasping concepts like inverse functions is critical. It enables the solving of complex equations and understanding of how functions behave under different compositions. By practicing function compositions and performing algebraic verification, students gain the groundwork needed for calculus.