Objective: This demo is designed to help students use graphical representations of functions to determine the domain and range. Level: This demo is appropriate for any course in which functions are discussed. It is especially useful for courses such as elementary algebra, precalculus, and calculus. Platform: The examples in the demo use Excel spreadsheets, Quicktime movies, animated gifs, and Java applets. The animated gifs run in a browser or in a utility such as Quicktime player. Quicktime player is required to play the movies. Instructor's Notes: An understanding of the various representations (numerical, graphical, symbolic, and tabular) of functions is of fundamental importance in mathematics. As early as the 35 grade band, the NCTM Standards [1] articulate the expectation that students will be able to "represent and analyze patterns and functions using words, tables, and graphs." More advanced grade bands also place a high importance on the various representations of functions. This expectation is echoed in the AP Calculus course description [2]. A recent study [3] suggested that not enough emphasis is placed upon the the graphical representation of functions and that students need to be trained in visual thinking before they get to calculus. In this demo, students are trained to look at values on the vertical axis when they are determining the range and values on the horizontal axis when determining the domain. This collection of examples was developed so that instructors may use animations and/or Excel spreadsheets to demonstrate how to use a graph to find the domain and range of a function. Excel spreadsheets can then be used by the students for individual or group assignments. Questions accompany the spreadsheet to guide the student's work. In this demo, animations and sliderdriven applications help students to make connections between the points on a graph of a function and the domain and range of the function. A typical type of textbook problem (see 45 in the references) is to consider a graph and to use the graph to determine the domain and range of a function. We consider several examples in order of increasing difficulty. Example 1: A Function Defined by a Formula Use the graph to find the domain and range of the function f(x) = sqrt(x).
A graphical approach to obtaining the domain is to trace the points on the graph of the function and plot the xcoordinates of the points on a horizontal axis. The animation below illustrates the process.
Notice that as the point moves along the graph of f, the xcoordinates are plotted on the horizontal axis. The arrow indicates that the graph of the domain continues. Thus, the domain of the function is . Similarly, if we trace the points on the graph of the function and plot the ycoordinates on a vertical axis, we will have a graphical depiction of the range of the function. This is illustrated below. The arrow indicates that the graph of the range continues.
The range of the function is . Example 2: A Function Defined by a Formula Find the domain and range of the reciprocal function, f(x) = 1/x.
As in the previous example we trace the graph. The xcoordinates of the points on the graph, when plotted on the horizontal axis give a graphical depiction of the domain.
Notice that the function is undefined at the origin. Thus the domain of the function is . Similarly, the ycoordinates of the points on the graph, when plotted on the vertical axis, give a graphical depiction of the range.
A comprehensive collection of examples of this type can be found by clicking here. Excel worksheets, animated gifs, and Quicktime movie files are available. Examples 3 and 4 illustrate visualizations of domain and range of piecewise functions. Example 3: An "Easy" Piecewise Function Find the domain and range of the function shown in the graph below.
Students often have difficulty with piecewise functions, their definitions, and graphs. However, obtaining the domain and range of the function from the graph follows the same procedure as for functions whose graph is determined from a particular formula. To visualize the domain, plot the xcoordinates of the points along the graph of the function on a horizontal axis. The domain is the interval [3,3].
To visualize the range, plot the ycoordinates of the points on the graph of the function on a vertical axis. The range is the interval [2,4]. Example 4. A "Harder" Piecewise Function Find the domain and range of the function whose graph is shown below.
Notice that this function has some jump discontinuities and also some endpoints that are not included. Obtain the domain by tracing the points on the graph and plot the xcoordinates of the points on a horizontal axis, as illustrated in the animation below. The domain is the set of points [6,0] U [1,2) U [3,6]. In a similar way, obtain the range by tracing the points on the graph and plot the ycoordinates of the points on a vertical axis, shown in the animation below. The range is the set [3,1] U [2,9]. It is important to point out that although there is an "open" point at coordinates (2,4), there is a point on the graph with ycoordinate 4 that "closes up" the range interval. A comprehensive collection of examples of this type can be found by clicking here. Excel worksheets, animated gifs, and Quicktime movie files are available. ADDITIONAL COMMENTS Recent studies by Mazur [6] suggest that demonstrations can be made more effective by asking students to predict the outcome, seeing the demo, and then articulating what they saw after the demonstration is performed. Thus, student involvement in the demo seems to be an important component in a demo's effectiveness as a teaching/learning tool. The interactive nature of the Excel spreadsheets and the questions that accompany the Excel spreadsheets will encourage student engagement. References
1. Principles and Standards for School
Mathematics, National Council of Teachers of Mathematics,
http://standards.nctm.org 4. David Cohen, Precalculus with UnitCircle Trigonometry, Third Edition, Brooks/Cole, 1998. 5. Ron Larson, Robert P. Hostetler, and Bruce H. Edwards, Precalculus: A Graphing Approach, Third Edition, Houghton Mifflin Company, 2001.
6. The Mazur Group, http://mazurwww.harvard.edu/research/detailspage.php?ed=1&rowid=10.
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LFR 9/21/04 Last updated 5/19/2006 DRH
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