The function is increasing on 2 3 and constant on 2

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Unformatted text preview: ons involving absolute value we make liberal use of Definition 2.4, as the next example illustrates. Example 2.2.1. Graph each of the following functions. Find the zeros of each function and the x- and y -intercepts of each graph, if any exist. From the graph, determine the domain and range of each function, list the intervals on which the function is increasing, decreasing, or constant, and find the relative and absolute extrema, if they exist. 1. f (x) = |x| 3. h(x) = |x| − 3 2. g (x) = |x − 3| 4. i(x) = 4 − 2|3x + 1| Solution. 1. To find the zeros of f , we set f (x) = 0. We get |x| = 0, which, by Theorem 2.1 gives us x = 0. Since the zeros of f are the x-coordinates of the x-intercepts of the graph of y = f (x), we get (0, 0) is our only x-intercept. To find the y -intercept, we set x = 0, and find y = f (0) = 0, so that (0, 0) is our y -intercept as well.1 With Section 2.1 under our belts, we can use Definition 2.4 to get f (x) = |x| = −x, if x < 0 x, if x ≥ 0 Hence, for x < 0, we are graphing the line y = −x; for x ≥ 0, we have the line y = x. Proceeding as we did in Section 1.7, we get 1 Actually, since functions can have at most one y -intercept...
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