Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Graph each finction in the standand viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. $$f(x)=x^{5} ;(-\infty, \infty)$$

Step-by-Step Solution

Answer

The function is increasing on \((-\infty, \infty)\).

Step 1: Understand the Function

The given function is \( f(x) = x^{5} \). This is a polynomial function of degree 5, where the leading coefficient is positive (1). It's an odd-degree polynomial function.

Step 2: Analyze the Behavior of the Function

Since \( f(x) = x^{5} \) has a positive leading coefficient, it will have endpoints going in opposite directions: as \( x \to -\infty \), \( f(x) \to -\infty \) and as \( x \to \infty \), \( f(x) \to \infty \). This implies the function is increasing as a whole.

Step 3: Graph the Function

Using a graphing calculator, enter \( f(x) = x^{5} \) and observe the graph in the standard viewing window. The graph should start from the bottom left and rise to the top right.

Step 4: Trace the Graph

Using the trace function on the calculator, move from left to right along the graph. Observe the values of \( f(x) \) as \( x \) increases to confirm the behavior of the function.

Step 5: Determine the Interval Behavior

Since you observed that as you trace from left to right the function is continuously rising, \( f(x) \) is increasing over the interval \((-\infty, \infty)\).

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool that allows us to visualize different kinds of mathematical functions. For a function like \( f(x) = x^5 \), which is a polynomial function of degree 5, a graphing calculator can help us understand its shape and behavior. To begin:

• First, input the function \( f(x) = x^5 \) into your graphing calculator. This will produce a visual representation of the function.
• Choose the standard viewing window, typically set to show \( x \) and \( y \) values within a range of -10 to 10. This range often provides a good overall view of many functions, including polynomials.
• Observe the graph that appears. You should see a curve that starts low on the left (bottom-left) and rises as it moves to the right (top-right).

The graphing calculator gives us a quick visual insight into the function, which is particularly useful for understanding complex functions at a glance.

Function Behavior

The behavior of a function refers to how the function behaves as \( x \) goes towards positive and negative infinity. For polynomial functions like \( f(x) = x^5 \), the degree and the leading coefficient significantly influence its behavior.

• In our example, \( f(x) = x^5 \) is an odd-degree polynomial with a leading coefficient of 1, which is positive.
• This means that as \( x \) approaches \(-\infty \), \( f(x) \) will decrease towards \(-\infty \). Conversely, as \( x \) approaches \( \infty \), \( f(x) \) will increase towards \( \infty \).

Understanding this end behavior helps us predict the overall direction of the graph. Odd-degree polynomials with positive leading coefficients will always have this opposing behavior at the ends of the graph.

Increasing Functions

An increasing function is one where, as you move from left to right along the graph, the function values rise. For \( f(x) = x^5 \), this can be examined using the trace feature on a graphing calculator.

• With the calculator, start tracing the graph from the far left. Watch how the \( y \)-values (or \( f(x) \) values) change as \( x \) increases.
• If the \( y \)-values continuously increase as you slide your trace across the screen from left to right, the function is increasing over that interval.
• In the specific case of \( f(x) = x^5 \), no matter where the trace starts or ends along the interval \((-\infty, \infty)\), the function will always be increasing.

Thus, for the interval \((-\infty, \infty)\), \( f(x) = x^5 \) is confirmed to be an increasing function, reflecting its overall behavior.