By R. Barnett, et. al.

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The x coordinate is the value we are looking for. We can use the INTERSECT command to find the coordinates of the intersection point in Figure 21. When we select the INTERSECT command, we are asked to make three choices: the first curve, the second curve, and a guess. When the desired equation is displayed at the top of the screen, press ENTER to select it (Figs. 22 and 23). It doesn’t matter which of the two graphs is designated as the first curve. qxd 11/22/2007 06:14 PM Page 12 pinnacle 110:MHIA064:mhbar3:SE:CH 01: 12 CHAPTER 1 FUNCTIONS, GRAPHS, AND MODELS To enter a guess, move the cursor close to the intersection point (Fig.

On the other hand, there are vertical lines that intersect the circle in Figure 1(b) in two points. For example, the vertical line through x ϭ 2 intersects the circle in the points (2, 2 13) and (2, Ϫ2 13) [Fig. 1(b)]. Thus, to the x value 2 there correspond two y values, 213 and Ϫ213. Consequently, the equation x2 ϩ y2 ϭ 16 does not define a function. These observations are generalized in Theorem 1. Z THEOREM 1 Vertical Line Test for a Function An equation defines a function if each vertical line in the rectangular coordinate system passes through at most one point on the graph of the equation.

The notation does not represent a product. It tells us that the function named f has independent variable x. f (x) is the value of the function f at x. 2(x) ϭ 2x is algebraic multiplication. qxd 11/22/2007 06:15 PM Page 31 pinnacle 110:MHIA064:mhbar3:SE:CH 01: S E C T I O N 1–2 Functions 31 Using function notation, f (3) is the output for the function f associated with the input 3. We find this range value by replacing x with 3 wherever x occurs in the function definition f(x) ϭ 2x ϩ 1 and evaluating the right side, f(3) ϭ 2 ؒ 3 ϩ 1 ϭ6ϩ1 ϭ7 The statement f (3) ϭ 7 indicates in a concise way that the function f assigns the range value 7 to the domain value 3 or, equivalently, that the ordered pair (3, 7) belongs to f.

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