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The cubed root function is a mathematical operation that finds the number which, when multiplied by itself three times, results in the original number. For example, the cubed root of 8 is 2, because \(2 \times 2 \times 2 = 8\). Unlike square roots, cubed roots can be taken for both positive and negative numbers. This is because taking the cube of a negative number will still yield a negative result, meaning \(\sqrt[3]{-8} = -2\), because \((-2) \times (-2) \times (-2) = -8\).
Understanding the behavior of the cubed root function gives insight into how it acts as the inverse of the cubic function \(f(x) = x^3\). In this case, the inverse function is \(f^{-1}(x) = \sqrt[3]{x}\).

• This means applying \(f\) and then \(f^{-1}\) to a number will give you back the original number, so \(f(f^{-1}(x)) = x\).
• The cubed root function covers all real numbers and is continuously increasing across its domain.

This feature makes it a crucial and well-behaved function for further mathematical analysis and application.

The Horizontal Line Test is a method used to determine whether a function has an inverse that can also be a function. The rule is simple: If any horizontal line drawn on the graph of the function crosses it more than once, then the function does not have an inverse that is also a function.
When analyzing the cubic function \(f(x) = x^3\), you can apply the horizontal line test to its graph.

• This graph passes the horizontal line test because any horizontal line you draw will intersect the graph at exactly one point.
• This implies that its inverse is a function.

Consequently, for the function \(y = \sqrt[3]{x}\), which is the inverse of \(f(x) = x^3\), the horizontal line test similarly verifies that this function preserves the unique mapping found in the original function, affirming its own validity as a function.

Function analysis involves evaluating both the original function and its inverse to ensure they conform to certain mathematical properties. It’s important to check whether the calculations for the inverse function make sense and whether the inverse behaves as expected.
For \(f(x) = x^3\), the steps to finding the inverse included:

• Replacing \(f(x)\) with \(y\), obtaining \(y = x^3\)
• Swapping \(x\) and \(y\) to get \(x = y^3\)
• Solving for \(y\) by taking the cubed root, which gives \(y = \sqrt[3]{x}\)

A successful function analysis checks these manipulations to ensure logical steps were followed, resulting in \(f^{-1}(x) = \sqrt[3]{x}\).
This analysis confirms the inverse is also smooth and continuous across all real numbers, reflecting stable input-output relationships.
Employing tests like the Horizontal Line Test during function analysis further verifies that the inverse function retains the necessary mathematical properties to be considered a function, guaranteeing that each element in the domain corresponds to exactly one element in the range.