Section 3.3 Piece Functions

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.

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

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

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

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}

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.

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

Graph ƒ(x) = [x] y x

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

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

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

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

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)}

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

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

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

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

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.

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.

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.

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

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.

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

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

Homework: pp. 123-125

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

►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

►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|

►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

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

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

■ Cumulative Review 37. Give the reference angles for the following angles: 117°, 201°, 295°, -47°.

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

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

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

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