Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Show that \(f\) and \(g\) are inverse functions algebraically. Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{3}, \quad g(x)=\sqrt[3]{x}$$

Step-by-Step Solution

Answer

The functions \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) are indeed inverse functions of each other. This is confirmed algebraically by finding that \(f(g(x)) = x\) and \(g(f(x)) = x\). In addition, a graphical interpretation shows that the graphs of both functions are reflected across the line \(y = x\), further supporting the inverse relationship.

Step 1: Algebraic Proof

Prove that \(f(g(x)) = x\) and \(g(f(x)) = x\). For \(f(g(x))\), we substitute \(g(x) = \sqrt[3]{x}\) into \(f(x)\): \(f(g(x)) = (\sqrt[3]{x})^3 = x\). For \(g(f(x))\), we substitute \(f(x) = x^3\) into \(g(x)\): \(g(f(x)) = \sqrt[3]{(x^3)} = x\). Therefore, \(f\) and \(g\) are inverse functions.

Step 2: Graphical Interpretation

By graphing both \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\), we can observe that the graphs are reflections of each other across the line \(y = x\). This reflection demonstrates that they are indeed inverses of each other. Be aware that you can't technically 'see' this in the JSON response, but it's crucial for visualizing the relation.

Step 3: Describing the Relationship

Both of the functions are cubic. The graph of \(f(x) = x^3\) ascends from left to right, which means it's an increasing function. On the contrary, the graph of its inverse function \(g(x) = \sqrt[3]{x}\) is a reflection of \(f(x)\) across the line \(y = x\). This means for every point on \(f(x)\) such as (a,b), there would be a corresponding point (b,a) on \(g(x)\).

Key Concepts

Algebraic Proof of Inverse Functions

In mathematics, proving that two functions are inverses of each other involves demonstrating that when you combine them, in both possible ways, they return the original value or input. This can be done algebraically for functions like \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\), where \(f\) and \(g\) should satisfy the equations, \(f(g(x)) = x\) and \(g(f(x)) = x\).
To show this:

• Substitute \(g(x) = \sqrt[3]{x}\) into \(f(x) = x^3\):

\[ f(g(x)) = (\sqrt[3]{x})^3 = x \]

• Switch the functions, substitute \(f(x) = x^3\) into \(g(x) = \sqrt[3]{x}\):

\[ g(f(x)) = \sqrt[3]{(x^3)} = x \] These operations verify that both functions return the original input \(x\). Thus, \(f\) and \(g\) are confirmed as inverse functions.
Understanding the algebraic proof of inverse functions is fundamental, as it provides the basis for predicting function behavior across different contexts.

Using Graphing Utilities for Inverse Function

Graphing utilities offer a practical visual aid for understanding functions and their inverses. When you plot \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) in the same graphing window, the utility will help you see how the two functions behave in relation to each other.
Here's how to interpret these graphs using a graphing utility:

• Input both \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) into the utility.
• Observe how the graphs are reflections of one another.

This reflection occurs across the line \(y = x\), confirming the inverse relationship visually. Unlike algebraic proof, this method gives you an instant, clear picture of their reciprocal nature.
Graphing utilities make learning about functions interactive. They're excellent tools for visually confirming mathematical relationships and gaining deeper insights into the nature of functions.

Graphical Interpretation of Inverse Functions

Seeing the graphs of inverse functions can illuminate our understanding of their characteristics. The functions \(f(x) = x^3\) and \(g(x) = \sqrt[3]{x}\) are closely related through their graphs.
Here's what you can observe:

• The graph of \(f(x) = x^3\) is a curve that moves upwards as it traverses from left to right, indicating it is increasing.
• The graph of \(g(x) = \sqrt[3]{x}\) reflects be the inverse, mirrored over \(y = x\). For every point \((a, b)\) on \(f(x)\), there exists a corresponding reflection \((b, a)\) on \(g(x)\).

This symmetry is a hallmark of inverse functions. Observing this in a graph helps solidify the concept: functions that are inverses of each other exhibit this mirror image relation across the line \(y = x\).
Interpreting this graphical transformation reinforces algebraic understanding and validates the inverse relationship. It’s one of the beautiful intersections of algebra and geometry where visual cues help clarify mathematical proofs.