Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The identity function \(I\) is the function whose input equals its output: \(I(x)=x\). If functions \(f\) and \(g\) have the property that \(f(g(x))=I(x)\) and \(g(f(x))=I(x)\), then \(f\) and \(g\) are inverse functions. For each function below, find the inverse function \(g(x)\) and verify that \(f(g(x))=I(x)\) and \(g(f(x))=I(x)\). (a) \(f(x)=6 x-3\) (b) \(f(x)=(x-3)^{3}\)

Step-by-Step Solution

Answer

The inverse of function \(f(x) = 6x - 3\) is \(g(x) = (x + 3)/6\). The inverse of function \(f(x) = (x-3)^{3}\) is \(g(x) = \sqrt[3]{x} + 3\). In both cases \(f(g(x)) = x = I(x)\) and \(g(f(x)) = x = I(x)\), verifying their inverse relationship.

Step 1: Solving function a: Finding the inverse of \(f(x)=6x-3\)

The given function is \(f(x) = 6x - 3\). To find its inverse, replace \(f(x)\) with \(y\): \(y = 6x - 3\). Swap \(x\) and \(y\): \(x = 6y - 3\). Solve for \(y\): \(y = (x + 3)/6\). Therefore, the inverse function \(g(x) = (x + 3)/6\).

Step 2: Solving function a: Verifying the inverses

To verify, substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = 6 * (x+3)/6 - 3 = x = I(x)\). Now, substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = ((6x - 3) + 3)/6 = x = I(x)\). Hence, \(f(x)\) and \(g(x)\) are indeed inverse functions.

Step 3: Solving function b: Finding the inverse of \(f(x)=(x-3)^{3}\)

The given function is \(f(x) = (x-3)^{3}\). To find its inverse, replace \(f(x)\) with \(y\): \(y = (x-3)^{3}\). Swap \(x\) and \(y\): \(x = (y-3)^{3}\). Solve for \(y\): \(y = \sqrt[3]{x} + 3\). Therefore, the inverse function \(g(x) = \sqrt[3]{x} + 3\).

Step 4: Solving function b: Verifying the inverses

To verify, substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = ((\sqrt[3]{x} + 3)-3)^{3} = x = I(x)\). Now, substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = \sqrt[3]{(x-3)^{3}} + 3 = x = I(x)\). Hence, \(f(x)\) and \(g(x)\) are definitely inverse functions.

Key Concepts

Identity Function

The identity function is a special function where the input is exactly equal to the output. In mathematical terms, it's expressed as \( I(x) = x \). This function serves as the baseline or the simplest form of a function. Whenever you compose a function with its inverse and end up with the identity function, you've hit a significant mathematical milestone. This shows that each function “undoes” the other, bringing back the original input.

When working with inverse functions, achieving the identity function after composition implies correctness. It confirms that the operations have successfully reversed each other.

Function Composition

Function composition involves combining two or more functions to create a new function. For any two functions \( f \) and \( g \), the composition \( f(g(x)) \) means you're applying \( g \) first and then applying \( f \) to the result. For functions to be inverses, two compositions need to yield the identity function:

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

In these compositions, each function should perfectly counteract the other.

This is a key step in confirming that you have correctly identified or calculated inverse functions.

Inverse Verification

Inverse verification is the process of checking whether two functions are truly inverses of each other. After deriving what you think is the inverse of a function, you must substitute back into the original function and vice versa. Mathematically, you must show that

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

These conditions assure us that the original function and its inverse truly "cancel" each other out, reverting back to the input.

In the example above, solving for \( f(x) = 6x - 3 \) and verifying the inverse \( g(x) = \frac{x + 3}{6} \) confirms correctness.

Solving Equations

Solving equations often comes into play when finding the inverse of a function. The steps involve setting the function equal to \( y \), swapping \( x \) and \( y \), and then isolating \( y \).

For example, to find the inverse of \( f(x) = (x - 3)^3 \), you first replace \( f(x) \) with \( y \) giving you \( y = (x - 3)^3 \). Then swap \( x \) and \( y \): \( x = (y - 3)^3 \). Finally, solve for \( y \) to get \( y = \sqrt[3]{x} + 3 \).

This process results in the inverse function, which can then be verified through inverse verification methods to ensure accuracy.