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**Piecewise-defined Functions**

y x O Ex. 1: x y – h/d h d d ... ½ x – 2, x > 2 (1, 3) f(x) = 3, x = 1 x –2 – y h/d d d d h –2x + 3, –2 ≤ x < 1 Domain: [–2, 1] (2, ∞) Range: (–1, ∞) Evaluate: a) f(0) = 3 b) f(1) = 3 c) f(2) = undefined y x O Ex. 2: x –3 –4 –5 ... y –4 –5 –6 ... h/d h d d ... x – 1, x < –3 (4, 1) f(x) = –2/3 x + 2, –3 ≤ x < 3 x – y h/d d d h x – 3, x = 4 Domain: (–∞, 3) {4} Range: (–∞, –4) (0, 4] Evaluate: a) f(–1) = 22/3 b) f(3) = undefined c) f(4) = 1

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**Piecewise-defined Functions (cont’d)**

y x O Ex. 3: x y –2 –5 ... h/d h d d ... –3/2 x + 4, x > 2 f(x) = x –3 – y –7 –5 – h/d h d d d d 2x – 1, –3 < x ≤ 2 Domain: (–3, ∞) Range: (–∞, 3] Evaluate: a) f(–2) = –5 b) f(2) = 3 c) f(4) = –2 y x O x –3 –2 – y h/d d d d d h Ex. 4: |x|, –3 ≤ x < 1 x y h/d d h f(x) = 3, 1 ≤ x < 2 (3, –2) x – 5, x = 3 Domain: [–3, 2) {3} Range: _{–2} [0, 3]_ Evaluate: a) f(–3) = 3 b) f(1.5) = 3 c) f(3) = –2

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**Piecewise-defined Functions (cont’d)**

y x O x … y … h/d h d d … Ex. 5: x, x > 1 x –2 – y h/d h d d d f(x) = x2, –2 < x ≤ 1 –2x – 7, x ≤ –2 x –2 –3 –4 … y –3 – … h/d d d d … Domain: (–∞, ∞) Range: [–3, ∞) Evaluate: a) f(4) = b) f(1) = c) f(–3) = –1 y x O Ex. 6: ½x + 3/2, x ≥ 1 1, –3 ≤ x < 1 x + 2, x < –3 f(x) = Domain: (–∞, ∞) Range: (–∞, –1) {1} [2, ∞) Increasing: (–∞, –3) (1, ∞) Decreasing: None

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**Piecewise-defined Functions (cont’d)**

y x O is in Ex. 7: –½x, x (2, ∞) x2, x (–1, 2] –2x – 1, x (–∞, –1) f(x) = Domain: (–∞, –1) (–1, ∞) Range: (–∞, –1) (0, ∞) Increasing: (0, 2) Decreasing: (–∞, –1) (–1, 0) (2, ∞) y x O Ex. 8: 2x – 6, x [3, ∞) |x|, x [–2, 3) x + 4, x (–∞, –2) f(x) = Domain: (–∞, ∞) Range: (–∞, ∞) Increasing: (–∞, –2) (0, 3) Decreasing: (–2, 0) (3, ∞) 2x – 6, x [3, ∞) |x|, x (–2, 3) x + 4, x (–∞, –2] OR f(x) =

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3.5 Continuity & End Behavior

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