1. Section 3.3
Piece Functions

2. Objectives:
1. To define and evaluate piece
functions.
2. To graph piece functions and
determine their domains and
ranges.
3. To introduce continuity of a function.

3. Definition
Piece functions are functions that requires two or more function rules to define them.

4. EXAMPLE 1 Evaluate f(0) and f(3)
for f(x) = .   
-3x + 2 if x  1
2x if x  1
f(0) = -3(0) + 2 = 2
f(3) = 23 = 8

5. EXAMPLE 2 Graph f(x) = . Give the  -3x + 2 if x  1 domain and range. 
-3x + 2 if x  1
2x if x  1

6. EXAMPLE 2 Graph f(x) = . Give the  -3x + 2 if x  1 domain and range. 
-3x + 2 if x  1
2x if x  1
D = {real numbers}
R = {y|y  -1}

7. Definition
A greatest integer function is a step function, written as ƒ(x) = [x], where ƒ(x) is the greatest integer less than or equal to x.

8. EXAMPLE 3 Find the set of ordered pairs described by the greatest integers function f(x) = [x] and the domain {-5, -3/2,
-3/4, 0, 1/4, 5/2}.
f(-5) = [-5] = -5
f(-3/2) = [-3/2] = -2
f(0) = [0] = 0
f(1/4) = [1/4] = 0
f(5/2) = [5/2] = 2

9. Graph ƒ(x) = [x] y x

10. f(x) = [x] = ï î í ì ... < £ 2 x 1 if - -1 -2
The rule for the greatest integer function can be written as a piece function.
f(x) = [x] = ï î í ì
...
< 2 x 1 if - -1 -2

11. Practice: Find f(2.75) for the function f(x) = [x].

12. Practice: Find f(-0.9) for the function f(x) = [x].

13. EXAMPLE 4 Find the function described by the function rule g(x) =
|2x – 3| for the domain {-4, -2, 0, 1, 2, 4}.
g(-4) = |2(-4) – 3| = |-11| = 11
g(-2) = |2(-2) – 3| = |-7| = 7
g(0) = |2(0) – 3| = |-3| = 3
g(1) = |2(1) – 3| = |-1| = 1
g(2) = |2(2) – 3| = |1| = 1
g(4) = |2(4) – 3| = |5| = 5

14. EXAMPLE 4 Find the function described by the function rule g(x) =
|2x – 3| for the domain {-4, -2, 0, 1, 2, 4}.
g = {(-4, 11), (-2, 7), (0, 3), (1, 1), (2, 1), (4, 5)}

15. Definition
Absolute value function The absolute value function is expressed as {(x, ƒ(x)) | ƒ(x) = |x|}.

16. Graph ƒ(x) = |x|   
x if x  0
-x if x  0
f(x) = |x| =

17. Plot the points (-3, 3), (-2, 2), (0, 0), (1, 1), (3, 3) and connect them to get the following.

18. EXAMPLE 5 Graph f(x) = |x| + 3. Give the domain and range.
{(-4, 7), (-2, 5), (0, 3), (1, 4), 3, 6)}

19. Translating Graphs
1. If x is replaced by x - a, where a  {real numbers}, the graph translates horizontally. If a > 0, the graph moves a units right, and if a < 0 (represented as x + a), it moves a units left.

20. Translating Graphs
2. If y, or ƒ(x), is replaced by y - b, where b {real numbers}, the graph translates vertically. If b > 0, the graph moves b units up, and if b < 0 (represented as y + b), it moves b units down.

21. Translating Graphs
3. If g(x) = -ƒ(x), then the functions ƒ(x) and g(x) are reflections of one
another across the x-axis.

22. Practice: Find the correct equation of the translated graph.
1. y = |x – 3| + 1
2. f(x) = |x + 3| + 1
3. y = |x + 1| - 3
4. f(x) = [x – 3] + 1

23. Continuous functions have no gaps, jumps, or holes
Continuous functions have no gaps, jumps, or holes. You can graph a continuous function without lifting your pencil from the paper.

24. EXAMPLE 6 Graph    2x + 3 if x  -2 g(x) = |x| if -1  x  1 .

25. EXAMPLE 6 Graph    2x + 3 if x  -2 g(x) = |x| if -1  x  1 .

26. Homework: pp

27. ►A. Exercises
Find the function described by the given rule and the domain {-4, -1/2, 0, 3/4, 2}.
3. h(x) = [x]

28. ►B. Exercises
Without graphing, tell where the graph of the given equation would translate from the standard position for that type of function.
11. f(x) = |x| - 7

29. ►B. Exercises
Without graphing, tell where the graph of the given equation would translate from the standard position for that type of function.
13. y = |x + 4|

30. ►B. Exercises
Without graphing, tell where the graph of the given equation would translate from the standard position for that type of function.
15. y = [x + 1] + 6

31. ►B. Exercises
Graph. Give the domain and range of each. Classify each as continuous or discountinuous.
23. g(x) = [x]

32. î í ì
►B. Exercises
Graph. Give the domain and range of each. Classify each as continuous or discountinuous.
29. f(x) =
4 otherwise
x2 if -2  x  2

33. ■ Cumulative Review
37. Give the reference angles for the following angles: 117°, 201°, 295°, °.

34. ■ Cumulative Review
38. Find the sine, cosine, and tangent of 2/3.

35. ■ Cumulative Review
39. Classify y = 7(0.85)x as exponential growth or decay.

36. ■ Cumulative Review Consider f(x) = –x² – 4x – 3.
40. Find f(-2) and f(-1/2).

37. ■ Cumulative Review Consider f(x) = –x² – 4x – 3.
41. Find the zeros of the function.