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Understanding Piecewise Functions: Definitions, Examples, and Graphing Techniques

This section focuses on piecewise functions, which consist of two or more function rules. It covers how to define, evaluate, and graph these functions while determining their domains and ranges. Key concepts include continuity, greatest integer functions, step functions, and absolute value functions. Examples illustrate evaluations and transformations, aiding in understanding how to analyze such functions. Homework exercises provide practice in finding function descriptions and graphing them while classifying continuity types.

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Understanding Piecewise Functions: Definitions, Examples, and Graphing Techniques

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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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