Problem 127 - Precalculus Essentials

Question

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$g(x)=\left|4-x^{2}\right|$$

Step-by-Step Solution

Answer

The function is increasing on the intervals [-2,0] and [0,2], decreasing on [-5,-2] and [2,5], and is not constant at any interval.

Step 1: Understand the Equation

The equation given is \(g(x) = |4 - x^{2}|\). This is an absolute value function with a parabolic base function \(x^{2}\) subtracted from 4.

Step 2: Graph the Function

Use a graphing utility to plot the function \(g(x) = |4 - x^{2}|\). In the graph, you'll notice the function forms a shape similar to an 'W' inside the rectangle [-5,5,1] by [-5,5,1]

Step 3: Determine Where the Function is Increasing

The function \(g(x)\) is increasing for the interval of \(-\sqrt{4}\) to 0 and from 0 to \(\sqrt{4}\), which are [-2,0] and [0,2] when converted to interval notation.

Step 4: Determine Where the Function is Decreasing

The function \(g(x)\) is decreasing for the interval of -5 to \(-\sqrt{4}\) and from \(\sqrt{4}\) to 5, which are [-5,-2] and [2,5] when converted to interval notation.

Step 5: Determine Where the Function is Constant

In the equation given, the function \(g(x)\) is not constant at any interval due to the nature of the absolute value function. It either increases or decreases but does not maintain a constant value over an interval.

Key Concepts

Understanding Absolute Value Functions

Absolute value functions are a fundamental concept in algebra that involve taking the absolute value of a variable expression, which means no matter what the input value is, the output is always non-negative.

\rFor example, the absolute value of both -3 and 3 is 3, often represented as |-3| = 3 and |3| = 3. Graphically, these functions create a 'V' shape when simple, and when combined with quadratic functions, as in our exercise with the function \( g(x) = |4 - x^2| \), it can create a 'W' shape.

The absolute value function can significantly alter the base function it is applied to. In the case of \( g(x) = |4 - x^2| \), the quadratic function \( x^2 \) is flipped over the x-axis for values of x that make \( 4 - x^2 \) negative, resulting in the 'W' shape observed on the graphing utility.

Identifying Increasing and Decreasing Intervals

An essential part of analyzing functions is to establish where they are increasing and decreasing. A function is increasing on an interval if, as x moves from left to right, the function's value rises. Conversely, a function is decreasing if the values fall.

\rIn the context of our absolute value function \( g(x) = |4 - x^2| \), we first identify the vertex of the parabola formed without the absolute value, which is at \( x = 0 \). From here, we see that as x moves away from 0 in either direction up to \( \pm\sqrt{4} \), the function value initially increases, creating the increasing intervals of [-2,0] and [0,2].

\rBeyond these points, as x continues to move away from 0, the value of the function decreases, resulting in decreasing intervals from [-5,-2] and [2,5]. It's important to note that there are no constant intervals for this function, as the value of \( g(x) \) is continually changing within the given range.

Graphing with a Utility

Graphing with a utility, such as a graphing calculator or computer software, is a crucial skill for students to visualize and understand complex functions. These tools allow for accurate and quick plotting of equations, making it easier to interpret the behavior of functions over different intervals.

\rWhen graphing the absolute value function \( g(x) = |4 - x^2| \), a graphing utility can be configured with a viewing rectangle, usually through setting the range for x and y values. For our exercise, we used the range [-5,5] for both x and y. The utility accurately represents the intervals where the function increases and decreases, helping to reinforce the algebraic determination of these intervals.

\rWhile graphing by hand is a valuable skill, graphing with a utility ensures precision, saves time, and often helps with identifying patterns and symmetries in functions that may not be immediately apparent otherwise. It's recommended for students to become familiar with the use of graphing utilities as part of their mathematical toolkit.