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Example 2.2.2. 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 132 Linear and Quadratic Functions 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|
x 2. g (x) = |x + 2| − |x − 3| + 1 Solution.
1. We ﬁrst note that, due to the fraction in the formula of f (x), x = 0. Thus the domain is
(−∞, 0) ∪ (0, ∞). To ﬁnd the zeros of f , we set f (x) = |x| = 0. This last equation implies
|x| = 0, which, from Theorem 2.1, implies x = 0. However, x = 0 is not in the domain of f ,
which means we have, in fact, no x-intercepts. For the same reason, we have no y -intercepts,
since f (0) is undeﬁned. Re-writing the absolute value in the function gives −x , if x < 0
f (x) =
= x , if x > 0
x −1, if x < 0
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