Problem 128 - 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. $$h(x)=|x-2|+|x+2|$$

Step-by-Step Solution

Answer

The function \( h(x) = |x-2| + |x+2| \) increases over the intervals \(-\infty < x < -2\) and \(2 < x < \infty\), decreases over the interval \(-2 < x < 2\) and is constant at \(x = -2\) and \(x = 2\).

Step 1: Plotting the function

Input the function \( h(x) = |x-2| + |x+2| \) into the graphing utility and plot it with the given viewing rectangle \([-5, 5, 1]\) by \([-5, 5, 1]\). The graph will indicate how the function behaves at different values of x.

Step 2: Analyze the graph

Examine the graph to identify the intervals at which the function increases or decreases. A function increases at a point if its value at that point is lesser than at any point to its right and decreases if its value at that point is greater than at any point to its right. A function becomes constant at a point when its value at that point equals the value at any point to its right.

Step 3: Identify the intervals

From the produced plot, the function increases over the intervals \(-\infty < x < -2\) and \(2 < x < \infty\). The function decreases over the interval \(-2 < x < 2\). The function is constant at \(x = -2\) and \(x = 2\).

Key Concepts

Piecewise Functions

Piecewise functions are special types of functions that have different expressions for different intervals of the input variable, usually defined by different conditions or domains. In simpler terms, a piecewise function is broken into "pieces," with each piece following its own rule or equation. This unique feature allows piecewise functions to model real-world scenarios where a relationship might change based on conditions, such as taxes changing with income brackets, or pricing changing based on quantity purchased.

For example, the function defined by - If when driving less than 60 miles per hour it's free, but above 60, it costs $5, then it is a piecewise function with two pieces. * $0$ if speed is less than 60
* $5$ if speed is above 60

When graphing piecewise functions using a graphing utility, you need to carefully input each piece of the function based on its corresponding conditions. Each section must reflect its individual equation over the correct interval, allowing you to visualize how the function behaves differently across the range of its inputs.

Intervals of Increase and Decrease

In mathematics, determining the intervals of increase and decrease helps us understand the behavior of functions. These intervals inform us where the function is rising or falling.

- **Increasing Interval:** A function is said to be increasing on an interval if the output of the function grows as the input increases. That means for any two values of x in the interval, if one value is smaller than the other, the function value is smaller for the smaller input.
- **Decreasing Interval:** Conversely, a function is decreasing on an interval if the output of the function reduces as the input increases. Here, the function value is larger for a smaller input.

Identifying these intervals typically involves looking at the slope of the function. If it’s positive, the function increases; if it’s negative, the function decreases. This provides crucial information for analysis and helps in applications such as maximizing efficiency or minimizing waste in various processes. By using graphing utilities, you can visually identify where these changes occur.

Absolute Value Functions

Absolute value functions are mathematical functions that describe the distance of a number from zero on the number line, regardless of direction. The absolute value is always non-negative. When you see an expression like \[ |x| \], it represents the absolute value of x, which is:- Equal to x if x is positive or zero - Equal to \[ -x \] if x is negative.

This means that absolute value functions often create a V-shaped graph when plotted. For the function \[ h(x) = |x-2| + |x+2| \], two absolute value expressions are involved, which contributes to its shape and characteristics. Here, the two translations cause the "V" shape to move or translate on the x-axis.

Graphing absolute value functions often involves evaluating points where the function reaches a minimum or changes direction. These critical points occur at the vertices of the "V" and help define where the function's slope changes direction, marking key points of decreasing and increasing intervals.