Problem 26 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=x^{5}, \quad g(x)=\sqrt[5]{x}\)

Step-by-Step Solution

Answer

The functions \(f(x) = x^5\) and \(g(x) = \sqrt[5]{x}\) are inverses.

Step 1: Use the Inverse Function Property

To demonstrate that two functions are inverses, we need to verify that their compositions return the input. This means checking that \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 2: Compute \(f(g(x))\)

First, calculate \(f(g(x))\). Substitute \(g(x) = \sqrt[5]{x}\) into \(f(x) = x^5\), yielding:\[ f(g(x)) = f(\sqrt[5]{x}) = (\sqrt[5]{x})^5 \]Using exponent rules, we have:\[ (\sqrt[5]{x})^5 = x^{1/5 \, imes \, 5} = x \]

Step 3: Compute \(g(f(x))\)

Now, calculate \(g(f(x))\). Substitute \(f(x) = x^5\) into \(g(x) = \sqrt[5]{x}\), yielding:\[ g(f(x)) = g(x^5) = \sqrt[5]{x^5} \]Using the properties of exponents and radicals, this becomes:\[ \sqrt[5]{x^5} = x^{5/5} = x \]

Step 4: Verify Both Conditions Hold

Since both conditions: \(f(g(x)) = x\) and \(g(f(x)) = x\) are satisfied, \(f(x) = x^5\) and \(g(x) = \sqrt[5]{x}\) are inverses of each other.

Key Concepts

Function Composition

Function composition involves taking the output of one function and using it as the input to another. Imagine you have two functions: one is baking a cake, and the other is frosting it. The result of the first process becomes the starting point for the second. Let’s apply this notion to the exercise where you have the functions \(f(x) = x^5\) and \(g(x) = \sqrt[5]{x}\).
The function composition can be written as \(f(g(x))\) or \(g(f(x))\). For two functions to be inverses, both compositions must return the original input \(x\).

• For \(f(g(x))\), plug the output of \(g(x)\) into \(f(x)\).
• For \(g(f(x))\), use the result from \(f(x)\) as the input for \(g(x)\).

This process checks if performing one operation and then its inverse brings you back to where you started.

Inverse Functions

Inverse functions undo the action of their counterpart. It's like rewinding a video to return to the beginning. If \(f(x)\) applies an action to \(x\), then \(g(x)\) reverses it. To confirm if two functions are inverses, their compositions must satisfy \(f(g(x)) = x\) and \(g(f(x)) = x\).
In simple terms, this means:

• \(f(g(x))\) should equal \(x\) when processed through \(f(x)\).
• \(g(f(x))\) should also equate to \(x\) through \(g(x)\).

For our functions \(f(x) = x^5\) and \(g(x) = \sqrt[5]{x}\), using the Inverse Function Property was the key step to prove their inverse nature. This property ensures that operations cancel each other out.

Exponent Rules

Exponent rules describe how to manipulate mathematical expressions involving powers. They are fundamental in showing how \(f(x)\) and \(g(x)\) interact as inverses. Take the expression \((\sqrt[5]{x})^5\): using exponent rules, it simplifies to \(x\) because:

• The exponent rule \(a^{m} \,\cdot\, a^{n} = a^{m+n}\) helps us combine bases with like exponents.
• Applying \((\sqrt[5]{x})^5 = x^{1/5 \,\cdot\, 5} = x\) simplifies back to \(x\).

Similarly, the expression \(\sqrt[5]{x^5}\) also simplifies to \(x\) using the rule that states a power raised to a reciprocal power equals one. Understanding these rules allows us to follow and verify the steps taken in each function composition, asserting the inverse relationship between our functions.