Problem 78 - College Algebra

Question

In Exercises \(78-79,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \(200 \mathrm{M} \text { OUT }]\) feature to show that \(f\) and g have identical end behavior. $$f(x)=x^{3}-6 x+1, \quad g(x)=x^{3}$$

Step-by-Step Solution

Answer

Both the functions \(f(x) = x^{3}-6x+1\) and \(g(x) = x^{3}\) have identical end behavior. As \(x\) approaches infinity or negative infinity, both functions approach infinity or negative infinity respectively.

Step 1: Graph the function \(f(x)\)

Use a graphing utility to graph the function \(f(x) = x^{3}-6x+1\). The method to do this would depend on the particular graphing utility, but it generally involves inputting the equation of the function and setting the range of \(x\) and \(y\) values to be displayed.

Step 2: Graph the function \(g(x)\)

Next, proceed to graph the function \(g(x) = x^{3}\) using the same graphing utility and on the same viewing rectangle as the function \(f(x)\) to allow for easier comparison.

Step 3: Use the '200M OUT] feature

Use the '200M OUT] feature on the graphing utility. This mode zooms out to show the end behavior of the functions. It helps to visualize where the functions go as \(x\) goes to positive or negative infinity.

Step 4: Compare the End Behaviors

Finally, analyze the end behavior of the two functions. From the graph, it would be observed that as \(x\) approaches infinity, both \(f(x)\) and \(g(x)\) approach infinity. Similarly, as \(x\) approaches negative infinity, \(f(x)\) and \(g(x)\) approach negative infinity.

Key Concepts

Understanding End Behavior in Polynomial Functions

When studying polynomial functions, understanding the end behavior is crucial. End behavior refers to the behavior of a function as the input values (denoted as \(x\)) approach extremely large positive or negative numbers. For polynomial functions, this behavior is predominantly determined by the leading term, which is the term with the highest degree.

• If the leading term has an even degree, both ends of the polynomial will head in the same direction, either up or down as \(x\) goes to positive or negative infinity.
• If the leading term has an odd degree, one end will rise while the other falls.

For the functions \(f(x) = x^3 - 6x + 1\) and \(g(x) = x^3\), both have a leading term of \(x^3\); thus, they both exhibit odd-degree behavior. This means as \(x\) approaches positive infinity, both functions will rise to positive infinity, and as \(x\) approaches negative infinity, both will fall to negative infinity.

Using a Graphing Utility for Polynomial Functions

A graphing utility is an invaluable tool when analyzing polynomial functions. This tool allows you to input equations and quickly plot them visually to assist with understanding their behavior, such as intersections and asymptotes.
Using a graphing utility:

• Input the function equation, such as \(f(x) = x^3 - 6x + 1\) or \(g(x) = x^3\).
• Set appropriate ranges for \(x\) and \(y\) to capture significant features of the graph.
• Overlay different functions within the same viewing frame for comparison.

For this particular exercise, using graphing utilities enables you to easily observe how \(f(x)\) and \(g(x)\) behave similarly in terms of end behavior.

Explaining Infinity Limits in Polynomial Functions

Infinity limits in the context of polynomial functions refer to the value that a polynomial function approaches as \(x\) becomes infinitely large or small. Limits are powerful in understanding what happens at the extremes of a function’s domain.
When examining \(f(x) = x^3 - 6x + 1\) and \(g(x) = x^3\), these functions both tend towards infinity in similar ways:

• As \(x\) approaches positive infinity, \(f(x)\) closely follows the behavior of \(x^3\), eventually reaching positive infinity, present due to the dominant \(x^3\) component.
• Similarly, as \(x\) approaches negative infinity, \(f(x)\) behaves as \(-\infty\), again driven by the cubic term.

By understanding and calculating limits, you can better predict how these functions behave without necessarily graphing them, making it a powerful analytical tool in understanding polynomial functions.