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Section 3.3 Piece Functions

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.

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Section 3.3 Piece Functions

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  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 1Evaluate 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 domain and range. -3x + 2 if x  1 2x if x  1   

  6. EXAMPLE 2 Graph f(x) = . Give the 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 3Find 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. y x Graph ƒ(x) = [x]

  10. ì ì ï ï ... if - 2 £ x < - 1 ï ï ï ï -2 if - 1 £ x < 0 ï ï í í -1 if 0 £ x < 1 ï ï f(x) = [x] = ï ï 0 if 1 £ x < 2 ï ï ï ï 1 î î ... The rule for the greatest integer function can be written as a piece function.

  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 4Find 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 4Find 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. x if x  0 -x if x  0    f(x) = |x| = Graph ƒ(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 5Graph f(x) = |x| + 3. Give the domain and range. f(x) = |x| + 3 {(-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. You can graph a continuous function without lifting your pencil from the paper.

  24. EXAMPLE 6Graph 2x + 3 if x  -2 g(x) = |x| if -1  x  1 . x3 if x  1   

  25. EXAMPLE 6Graph 2x + 3 if x  -2 g(x) = |x| if -1  x  1 . x3 if x  1   

  26. Homework: pp. 123-125

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

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

  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.

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