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Answer:
a) 9
b) 1
c) -1
d) -b² +1
domain: (-∞, ∞)
range: (-∞, 1] ∪ [3, ∞)
Explanation:
When you evaluate a piecewise function, the first step is to determine what piece is applicable. Then, you fill in the argument value and do the arithmetic.
a) -6 < 2, so the first piece applies. f(-6) = |-6| +3 = 9
b) 2 = 2, so the second piece applies. f(2) = -2 +3 = 1
c) 4 > 2, so the second piece applies. f(4) = -4 +3 = -1
d) b² +2 ≥ 2, so the second piece applies. f(b² +2) = -(b² +2) +3 = -b² +1
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The function is defined for all values of x, so the domain is (-∞, ∞).
The minimum value of the first piece is +3. The maximum value of the second piece is f(2) = 1. So, values of y between 1 and 3 are not part of the range.
The range is (-∞, 1] ∪ [3, ∞).
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Additional comment
The square of a real number can never be negative, so b²+2 cannot be less than 2.
