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Chapter 1 | Functions and Graphs
Example 1.11 Working with the Absolute Value Function Find the domain and range of the function f ( x ) =2 | x −3 | +4.
Solution Since the absolute value function is defined for all real numbers, the domain of this function is (−∞, ∞). Since | x −3 | ≥0 for all x , the function f ( x ) =2 | x −3 | +4≥4. Therefore, the range is, at most, the set ⎧ ⎩ ⎨ y | y ≥4 ⎫ ⎭ ⎬ . To see that the range is, in fact, this whole set, we need to show that for y ≥4 there exists a real number x such that 2 | x −3 | +4= y . A real number x satisfies this equation as long as | x −3| = 1 2 ( y −4). Since y ≥4, we know y −4≥0, and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore, | x −3| = ⎧ ⎩ ⎨ −( x −3) if x <3 x −3 if x ≥3 .
Therefore, we see there are two solutions:
x = ± 1
y −4)+3.
2 (
⎨ y | y ≥4 ⎫ ⎭ ⎬ .
The range of this function is ⎧ ⎩
For the function f ( x ) = | x +2 | −4, find the domain and range.
1.8
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