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Unformatted text preview: Chapter 2 Review of Simple Functions In this chapter we review a few basic concepts related to functions, and introduce the simplest family of functions that have interesting, nontrivial properties: the power functions. We explore geometric and graphical properties of these, commenting on special attributes that will become important in later study. Using the power functions as basic building blocks, we construct the family of polynomials, and investigate how their features are inherited from the underlying behaviour of power functions. Here, we begin to develop a few important curve-sketching skills. We end the chapter with applications of these concepts to examples from biology. 2.1 What is a function A function is just a way of expressing a special relationship between a value we consider as the input ( x ) value and an associated output ( y ) value. We write this relationship in the form y = f ( x ) to indicate that y depends on x . The only constraint on this relationship is that, for every value of x we can get at most one value of y . This is equivalent to the vertical line property : the graph of a function can intersect a vertical line at most at one point. The set of all allowable x values is called the domain of the function, and the set of all resulting values of y are the range . Naturally, we will not always use the symbols x and y to represent independent and dependent variables. For example, the relationship V = 4 3 r 3 expresses a functional connection between the radius, r , and the volume, V , of a sphere. We say in such a case that V is a function of r . All the sketches shown in Figure 2.1 are valid functions. The first is merely a collection of points, x values and associated y values, the second a histogram. The third sketch is here meant to represent the collection of smooth continuous functions, and these are the variety of interest to us here in the study of calculus. On the other hand, the example shown in Figure 2.2 is not the graph of a function. We see that a vertical line intersects this curve at more than one point. This is not permitted, since as we already said, a given value of x should have only one corresponding values of y . v.2005.1 - September 23, 2009 1 Math 102 Notes Chapter 2 x x x y y y Figure 2.1: All the examples above represent functions. x y Figure 2.2: The above elliptical curve cannot be the graph of a function. The vertical line (shown dashed) intersects the graph at more than one point: This means that a given value of x corresponds to too many values of y . If we restrict ourselves to the top part of the ellipse only (or the bottom part only), then we can create a function which has the corresponding graph....
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This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at The University of British Columbia.
- Winter '09