Chapter 1 | Functions and Graphs
19
Solution a.
0
0.5
1.5
2.5
1
2
t
s ( t )
96
100
84
64
36
0
Table 1.3 Height s as a Function of Time t Since the ball hits the ground when t =2.5, the domain of this function is the interval [0, 2.5]. b.
Note that for this function and the function f ( x ) =−4 x +2 graphed in Figure 1.9 , the values of f ( x ) are getting smaller as x is getting larger. A function with this property is said to be decreasing. On the other hand, for the function f ( x ) = x +3+1 graphed in Figure 1.10 , the values of f ( x ) are getting larger as the values of x are getting larger. A function with this property is said to be increasing. It is important to note, however, that a function can be increasing on some interval or intervals and decreasing over a different interval or intervals. For example, using our temperature function in Figure 1.6 , we can see that the function is decreasing on the interval (0, 4), increasing on the interval (4, 14), and then decreasing on the interval (14, 23). We make the idea of a function increasing or decreasing over a particular interval more precise in the next definition.
Definition We say that a function f is increasing on the interval I if for all x 1 , x 2 ∈ I , f ( x 1 ) ≤ f ( x 2 )when x 1 < x 2 . We say f is strictly increasing on the interval I if for all x 1 , x 2 ∈ I ,
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