1. Piecewise-defined Functionsy 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

2. 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

3. 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

4. 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) =