Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse function. Then, graph the function and its inverse. \(f(x)=x^{3}-1\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{x + 1}\).

Step 1: Define the Function and Replace Notation

The original function provided is \(f(x) = x^3 - 1\). To find the inverse, first replace \(f(x)\) with \(y\), so the equation becomes \(y = x^3 - 1\).

Step 2: Solve for x

Next, solve for \(x\) in terms of \(y\). Start by adding 1 to both sides: \(y + 1 = x^3\). Then, take the cube root of both sides to isolate \(x\): \(x = \sqrt[3]{y + 1}\).

Step 3: Interchange Variables

For the inverse function, switch the roles of \(x\) and \(y\). This gives us the inverse function \(f^{-1}(x) = \sqrt[3]{x + 1}\).

Step 4: Confirm Inverse Function

To ensure \(f(x)\) and \(f^{-1}(x)\) are correct, check that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). For example, substitute \(f^{-1}(x) = \sqrt[3]{x + 1}\) back into \(f(x)\): \(f(\sqrt[3]{x + 1}) = (\sqrt[3]{x + 1})^3 - 1 = x + 1 - 1 = x\), confirming the correctness.

Step 5: Graph the Functions

Graph \(f(x) = x^3 - 1\) and its inverse \(f^{-1}(x) = \sqrt[3]{x + 1}\). The graph of the function and its inverse will be symmetric with respect to the line \(y = x\). Plot the curve for \(f(x)\) by testing values of \(x\) and obtaining corresponding \(y\) values, then perform the same for \(f^{-1}(x)\).

Key Concepts

Graphing Functions

Graphing functions helps us visualize the behavior of mathematical equations. For example, with the function \(f(x) = x^3 - 1\), we can see how values change when \(x\) is altered. Graphing doesn't just show us one value for one \(x\) but all possible values and their interactions over a range.When graphing, consider:

• Choosing points wisely: Test a variety of \(x\) values, including negatives, zero, and positives.
• Determine corresponding \(y\) values: Use the function to calculate this.
• Plot these points: Connect them smoothly to see overall trends.

Remember, the graph for inverse functions like \(f^{-1}(x)\) is mirrored around the line \(y = x\). This means for every point \((a, b)\) on \(f(x)\), \(f^{-1}(x)\) will have the point \((b, a)\). With practice, these visual representations will enhance understanding.

Function Notation

Function notation is the way we write functions for clear understanding. Instead of writing equations like \(y = x^3 - 1\), we write \(f(x) = x^3 - 1\). Here, \(f(x)\) signifies the function using \(x\) to determine \(y\).Why use function notation?

• It clearly identifies what you are working with, reducing confusion.
• Makes it easier to express complex ideas like inverse functions.
• Function notation works with any variable, not just \(x\).

In the provided exercise, the original function in notation form is \(f(x)\), and the inverse was expressed as \(f^{-1}(x)\), making it easy to differentiate between them. This format is crucial, especially in advanced math and its applications.

Checking Inverses

To confirm whether two functions are inverses, you need to perform specific checks. For \(f(x)\) and \(f^{-1}(x)\), you need to validate that applying one after the other returns the original input variable.Here's how to check inverses:

• Calculate \(f(f^{-1}(x)) = x\): Substituting the inverse into the original function should return \(x\).
• Calculate \(f^{-1}(f(x)) = x\): Substitute the original function into the inverse, you should again return \(x\).
• If both hold true, you've correctly found the inverse!

In our solution, these steps showed our functions \(f(x) = x^3 - 1\) and \(f^{-1}(x) = \sqrt[3]{x + 1}\) were correctly paired inverses—all this guarantees accuracy.

Cube Root Function

The cube root function, symbolized as \(\sqrt[3]{x}\), is central to solving equations like the inverse of \(x^3 - 1\). Unlike square roots, cube roots can evaluate both positive and negative numbers. Key properties of cube root functions:

• Every real number has exactly one real cube root.
• The function \(y = \sqrt[3]{x}\) is defined for all real \(x\).
• Graphically, the cube root function passes through the origin and is symmetric about the origin.

In our exercise, using \(\sqrt[3]{x + 1}\) allowed us to find the inverse of \(x^3 - 1\), demonstrating this function’s utility for solving equations that involve cubed terms. Understanding cube roots, therefore, is essential for diverse mathematical tasks.