1.1 dom,ran,cost,increase,formula1

1.1 dom,ran,cost,increase,formula1 - 12 | CHAPTER 1...

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It’s helpful to think of a function as a machine (see Figure 2). If is in the domain of the function then when enters the machine, it’s accepted as an input and the machine produces an output according to the rule of the function. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled ( or ) and enter the input x . If , then is not in the domain of this function; that is, is not an acceptable input, and the calculator will indi- cate an error. If , then an approximation to will appear in the display. Thus the key on your calculator is not quite the same as the exact mathematical function defined by . Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of to an element of . The arrow indicates that is associated with is associated with , and so on. The most common method for visualizing a function is its graph. If is a function with domain , then its graph is the set of ordered pairs (Notice that these are input-output pairs.) In other words, the graph of consists of all points in the coordinate plane such that and is in the domain of . The graph of a function gives us a useful picture of the behavior or “life history” of a function. Since the -coordinate of any point on the graph is , we can read the value of from the graph as being the height of the graph above the point (see Figure 4). The graph of also allows us to picture the domain of on the -axis and its range on the -axis as in Figure 5. EXAMPLE 1 The graph of a function is shown in Figure 6. (a) Find the values of and . (b) What are the domain and range of ? SOLUTION (a) We see from Figure 6 that the point lies on the graph of , so the value of at 1 is . (In other words, the point on the graph that lies above x 1 is 3 units above the x -axis.) When x 5, the graph lies about 0.7 unit below the x -axis, so we estimate that . (b) We see that is defined when , so the domain of is the closed inter- val . Notice that takes on all values from 2 to 4, so the range of is M y 2 y 4 2, 4 f f 0, 7 f 0 x 7 f x f 5 0.7 f 1 3 f f 1, 3 f f 5 f 1 f 0 y ƒ( x ) domain range FIGURE 4 {x, ƒ} ƒ f(1) f(2) 0 1 2 x FIGURE 5 x y x y y x f f x f x y f x x , y y f f x y f x x , y f x , f x x D D f a f a x , f x E D f x s x f s x s x x 0 x x x 0 s x s f x x f , x 12 |||| CHAPTER 1 FUNCTIONS AND MODELS FIGURE 2 Machine diagram for a function ƒ x (input) ƒ (output) f f D E ƒ f(a) a x FIGURE 3 Arrow diagram for ƒ FIGURE 6 x y 0 1 1 N The notation for intervals is given in Appendix A.

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EXAMPLE 2 Sketch the graph and find the domain and range of each function.
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