Problem 14 - Precalculus with Limits

Question

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = x^5\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = x^5\) is \(f^{-1}(x) = \sqrt[5]{x}\), and it has been verified that the properties \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) both hold.

Step 1: Find the Inverse Function

To find the inverse of a function, we need to switch \(x\) and \(y\) and then solve for \(y\). Start with the function \(f(x) = x^5\). Switch \(x\) and \(y\) to get \(x = y^5\). Now solve for \(y\) to find the inverse function. We find \(y = \sqrt[5]{x}\), so \(f^{-1}(x) = \sqrt[5]{x}\).

Step 2: Verify \(f(f^{-1}(x)) = x\)

Now we substitute \(f^{-1}(x)\) into \(f(x)\) to verify the property \(f(f^{-1}(x)) = x\). \(f(f^{-1}(x)) = (\sqrt[5]{x})^5 = x\). So the property \(f(f^{-1}(x)) = x\) holds.

Step 3: Verify \(f^{-1}(f(x)) = x\)

Finally, substitute \(f(x)\) into \(f^{-1}(x)\) to verify the property \(f^{-1}(f(x)) = x\). \(f^{-1}(f(x)) = \sqrt[5]{(x^5)} = x\). So the property \(f^{-1}(f(x)) = x\) also holds.

Key Concepts

Function Verification

When dealing with functions and their inverses, function verification is a crucial step. This process ensures that two functions, say a function and its inverse, are accurately defined. Verified functions guarantee that their inverse correctly undoes the work done by the original function.

In this context, if we have a function \( f(x) = x^5 \), we want to ensure that its inverse function indeed works properly. Verification requires two critical properties to be true:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

Each of these properties means that plugging the output of one function into its inverse should yield the original input value.

Performing these calculations confirms that our inverse calculations are correct and have been performed properly.

Property of Inverse Functions

The essence of inverse functions lies in their unique characteristics. An inverse function essentially "reverses" the operation done by the original function.

Looking at our function \( f(x) = x^5 \), its inverse is ascertained by switching \( x \) and \( y \), giving us \( x = y^5 \), and then solving for \( y \). This leads to the inverse function \( f^{-1}(x) = \sqrt[5]{x} \).
The key property of inverse functions is that applying the function and then its inverse (or vice versa) results in the initial value. So,

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

This property is the hallmark of inverse functions and assures us that the operations conducted by the original and inverse functions cancel each other out.

Solving for Inverse

To solve for the inverse of a function, a methodical approach is necessary. Let's explain with \( f(x) = x^5 \):

• **Switch the variables**: Replace \( f(x) = y \) with \( x = y^5 \) to indicate that we are now finding \( f^{-1}(x) \).
• **Solve for \( y \)**: Rearrange \( x = y^5 \) to isolate \( y \), by taking the fifth root of both sides, thus \( y = \sqrt[5]{x} \).
• **Express the inverse**: The inverse function is \( f^{-1}(x) = \sqrt[5]{x} \).

These steps ensure that the inverse is determined correctly, maintaining the symmetry between the function and its inverse. The ability to reverse the operations of the original function is pivotal in doing things like finding inverse operations in algebra and solving real-world problems.