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FUNCTIONS AND MODELS
1.1 Four Ways to Represent a Function 1. The functions () = +
√ √ 2 − and () = + 2 − give exactly the same output values for every input value, so
and are equal. 2. () =
( − 1) 2 − = = for − 1 6= 0, so and [where () = ] are not equal because (1) is undefined and −1 −1
(1) = 1. 3. (a) The point (−2 2) lies on the graph of , so (−2) = 2. Similarly, (0) = −2, (2) = 1, and (3) 25.
(b) Only the point (−4 3) on the graph has a value of 3, so the only value of for which () = 3 is −4. (c) The function outputs () are never greater than 3, so () ≤ 3 for the entire domain of the function. Thus, () ≤ 3 for −4 ≤ ≤ 4 (or, equivalently, on the interval [−4 4]). (d) The domain consists of all values on the graph of : { | −4 ≤ ≤ 4} = [−4 4]. The range of consists of all the values on the graph of : { | −2 ≤ ≤ 3} = [−2 3]. (e) For any 1 2 in the interval [0 2], we have (1 ) (2 ). [The graph rises from (0 −2) to (2 1).] Thus, () is increasing on [0 2]. 4. (a) From the graph, we have (−4) = −2 and (3) = 4.
(b) Since (−3) = −1 and (−3) = 2, or by observing that the graph of is above the graph of at = −3, (−3) is larger than (−3). (c) The graphs of and intersect at = −2 and = 2, so () = () at these two values of . (d) The graph of lies below or on the graph of for −4 ≤ ≤ −2 and for 2 ≤ ≤ 3. Thus, the intervals on which () ≤ () are [−4 −2] and [2 3]. (e) () = −1 is equivalent to = −1, and the points on the graph of with values of −1 are (−3 −1) and (4 −1), so the solution of the equation () = −1 is = −3 or = 4. (f) For any 1 2 in the interval [−4 0], we have (1 ) (2 ). Thus, () is decreasing on [−4 0]. (g) The domain of is { | −4 ≤ ≤ 4} = [−4 4]. The range of is { | −2 ≤ ≤ 3} = [−2 3]. (h) The domain of is { | −4 ≤ ≤ 3} = [−4 3]. Estimating the lowest point of the graph of as having coordinates (0 05), the range of is approximately { | 05 ≤ ≤ 4} = [05 4]. 5. From Figure 1 in the text, the lowest point occurs at about ( ) = (12 −85). The highest point occurs at about (17 115).
Thus, the range of the vertical ground acceleration is −85 ≤ ≤ 115. Written in interval notation, the range is [−85 115]. c 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. °
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