Problem 86 - College Algebra

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=\sqrt[3]{x}-2, g(x)=(x+2)^{3} $$

Step-by-Step Solution

Answer

By visually inspecting the graph, one can conclude whether \(f\) and \(g\) are inverses of each other by checking if the graph of \(g(x)\) is the reflection of the graph of \(f(x)\) on the line \(y=x\).

Step 1: Graph the functions

Input the equations \(f(x)=\sqrt[3]{x}-2\) and \(g(x)=(x+2)^{3}\) into a graphing utility. The graph of \(f\) should be a cubic root curve shifted two units down, and the graph of \(g\) should be a cubic curve shifted two units to the left.

Step 2: Graph the line \(y=x\)

Next, put the equation \(y=x\) into the graphing utility to see a straight line. This will be used as the mirror axis for checking symmetry between the two functions. If \(f\) and \(g\) are inverse functions, the graph of \(g(x)\) should be a reflection of the graph of \(f(x)\) across the line \(y=x\).

Step 3: Analyzing the Graphs

Visually determine if the graph of \(g(x)\) is the reflection of the graph of \(f(x)\) across the line \(y=x\). If they are reflections of each other across the line \(y=x\), then \(f\) and \(g\) are inverses.

Key Concepts

Graphing Utility

When exploring functions, a graphing utility is a powerful tool that comes into play. It helps visualize mathematical concepts and confirms their properties graphically. For example, when working with the functions f(x) = \( \sqrt[3]{x} - 2 \) and g(x) = \( (x + 2)^3 \) as in the exercise provided, a graphing utility will allow students to input these equations and instantly see their respective graphs.

This visual representation is crucial, especially when comparing two functions to determine if they are inverses. By seeing the two functions plotted within the same viewing rectangle, along with the line y = x, students can look for the symmetry that signifies an inverse relationship. A good graphing utility not only plots the functions accurately but also offers features like zooming and shifting the viewing window, which are essential for a thorough analysis.

Cubic Root Function

A cubic root function is expressed mathematically as f(x) = \( \sqrt[3]{x} \) and represents the inverse operation of raising a number to the power of three. When you graph a basic cubic root function, it resembles a sideways 'S' expanding outwards in both the positive and negative directions of the x-axis.

In the exercise, the cubic root function is modified to f(x) = \( \sqrt[3]{x} - 2 \), indicating a downward shift of two units from the original function. Understanding transformations of parent functions, like shifting, is essential when graphing and can aid in predicting the shape and position of the modified graph before utilizing a graphing tool. Notably, cubic root functions are odd, which means they are symmetrical about the origin, a characteristic that is preserved even after vertical or horizontal shifts.

Reflection Across the Line Y=X

The concept of reflection across the line y=x is a key aspect of understanding inverse functions. This line acts as a mirror, and if two functions are inverses of each other, their graphs will reflect across this line. Visually, you can think of the line y = x as a diagonal line that bisects the first and third quadrants of a coordinate plane.

In the context of the exercise, graphing the line y = x alongside the functions f(x) and g(x) allows students to verify their inverse relationship by observing symmetry. If for every point on f(x) there is a corresponding point on g(x) that seems to mirror it across the line y = x, it illustrates that f(x) and g(x) are indeed inverses. Remember, for any point (a, b) on the graph of f(x), there should be a point (b, a) on g(x) to satisfy the condition of being inverse functions.