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Chapter 1: Problem 36

Sketch the graph of the given function, evaluate the given expressions, and then use technology to duplicate the graphs. Give the technology formula. \(f(x)=\left\\{\begin{array}{ll}-x & \text { if }-1

Short Answer

Expert verified
The graph consists of three line segments: a decreasing line from (-1, undefined) to (0, 0) with open dot at (-1, undefined), an increasing line from (0, -2) to (2, 0) with solid dots at (0, -2) and (2, 0), and finally a decreasing line from (2, -2) to (4, -4) with an open dot at (2, -2) and a solid dot at (4, -4). The function evaluations are: \(f(0) = -2\), \(f(1) = -1\), \(f(2) = 0\), and \(f(3) = -3\). The technology formula to enter is: Piecewise[-x, -1 < x <= 0, x - 2, 0 <= x <= 2, -x, 2 < x <= 4].

Step by step solution

01

Identify the different cases of the piecewise function

We have a piecewise function with 3 different cases: 1. -x for -1 < x 鈮� 0 2. x - 2 for 0 鈮� x 鈮� 2 3. -x for 2 < x 鈮� 4
02

Sketch the graph

To graph the function, we handle the cases one by one and plot them within their respective intervals: 1. For the first case, -x, we note that we are in the interval -1 < x 鈮� 0. We find the endpoints of the interval by substituting the x-values, e.g., if we plug in x = -1 and x = 0, we get undefined value and -0 (0), respectively. The graph will be a decreasing line segment from point (-1, undefined) to (0, 0). Since -1 is not included in the domain, we put an open dot on (-1, undefined). 2. For the second case, x - 2, we are in the interval from 0 鈮� x 鈮� 2. Substitute x = 0 and x = 2, we get -2 and 0, respectively. The graph will be an increasing line segment from point (0, -2) to (2, 0). Since 0 and 2 are included in the domain, we put a solid dot on (0, -2) and (2, 0). 3. For the third case, -x, we are in the interval 2 < x 鈮� 4. Substitute x = 2 and x = 4, we get -2 and -4, respectively. The graph will be a decreasing line segment from point (2, -2) to (4, -4). Since 4 is included in the domain, we put an open dot on (2, -2) and a solid dot on (4, -4).
03

Evaluate the given expressions

a. To find f(0), we see that 0 is included in the second case, so we'd use x - 2: f(0) = 0 - 2 = -2. b. To find f(1), we see that 1 is included in the second case, so we'd use x - 2: f(1) = 1 - 2 = -1. c. To find f(2), we see that 2 is included in the second case, so we'd use x - 2: f(2) = 2 - 2 = 0. d. To find f(3), we see that 3 is included in the third case, so we'd use -x: f(3) = -3.
04

Provide the technology formula

For technology, such as graphing calculator or software like Desmos, to graph this function, you can enter as follows: Piecewise[-x, -1 < x <= 0, x - 2, 0 <= x <= 2, -x, 2 < x <= 4]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Piecewise Functions
Graphing piecewise functions can be a fun and insightful way to explore different behaviors within a single function. A piecewise function is defined by different expressions over different intervals. To graph these functions, you need to tackle each case separately and plot them on their respective intervals on the coordinate plane.

In the example function \(f(x)\), there are three distinct cases:
  • \(-x\) for \(-1 < x \leq 0\)
  • \(x - 2\) for \(0 \leq x \leq 2\)
  • \(-x\) for \(2 < x \leq 4\)
When sketching, draw each segment carefully, paying special attention to open and closed dots, which indicate if endpoint values are included or excluded.

A good approach is to plot the endpoint values by substituting the interval limits into the function. This shows where each section starts and ends. By following the guidelines for plotting and interpreting each segment, you'll have a clear visual representation of the piecewise function.
Evaluating Piecewise Functions
Evaluating piecewise functions simply means finding the value of the function at specific points. This involves determining which expression of the piecewise function to use based on the given input.

For instance, with \(f(x)\) as given, each input must be checked against the intervals:
  • For \(f(0)\), use \(x - 2\) because \(0\) falls in the interval \(0 \leq x \leq 2\). Thus, \(f(0) = -2\).
  • For \(f(1)\), use \(x - 2\) because \(1\) also falls within \(0 \leq x \leq 2\). Thus, \(f(1) = -1\).
  • For \(f(2)\), use \(x - 2\) since it falls at the edge of \(0 \leq x \leq 2\). Thus, \(f(2) = 0\).
  • For \(f(3)\), use \(-x\) because \(3\) lies in \(2 < x \leq 4\). Thus, \(f(3) = -3\).
These calculations clarify how the function behaves at specific points, grounding your understanding of its overall structure.
Using Technology in Mathematics
Utilizing technology in mathematics offers exciting opportunities to visualize and verify complex concepts like piecewise functions. Tools like graphing calculators or software such as Desmos can help you visualize these functions more dynamically.

With technology, you can easily input piecewise functions using a specific syntax. For example, in Desmos, you might enter the function as:
  • Piecewise[-x, -1 < x <= 0, x - 2, 0 <= x <= 2, -x, 2 < x <= 4]
These inputs can generate clear, precise graphs that visually convey the transitions between different function rules.

Technology not only assists in sketching but also in evaluating functions at specific points, checking your work, and providing a deeper understanding of how mathematical concepts play out visually.

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Most popular questions from this chapter

Find a linear equation whose graph is the straight line with the given properties. Through \((2,-4)\) and \((1,1)\)

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