1. PreCalculus Section 1-3 Graphs of Functions

2. Objectives Find the domain and range of functions from a graph. Use the vertical line test for functions. Determine intervals on which functions are increasing, decreasing, or constant. Determine relative maximum and relative minimum values of functions. Identify and graph step functions and other piecewise defined functions. Identify even and odd functions.

3. The Graph of a Function The graph of a function f is the collection of ordered pairs ( x, f(x) ) such that x is in the domain of f. The geometric interpretations of x and f(x) are; –x : the directed distance from the y -axis –f(x) : the directed distance from the x -axis The domain is {x| -2 ≤ x<5} The range is {y| -4 ≤ y ≤ 4}. f(x) x

4. x y 4 -4 The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists. The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. Example: Find the domain and range of f (x) = x 2 – 2x – 2 by investigating its graph. Domain = all real numbers Range The domain is the real numbers. The range is { y: y ≥ -3} or [-3, + ). Domain & Range

5. Example: Find the domain and range. x y 11 22 33 44 123 4 22 33 44 1 2 3 4 Domain: {-3, -2, -1, 0, 1, 2} Range: {-2, 0, 1, 2, 4} Domain & Range

6. Examples: Domain and Range from a Graph Domain Range Domain Calculate the Max. & Min. ∴ R = -6 ≤ y ≤ 6 or [-6, 6] Calculate the zeros ∴ D = -3 ≤ x ≤ 3 or [-3, 3] The parabola continues to ∞ and covers the entire x-axis ∴ D = ℝ or (- ∞, ∞ ) Calculate the Min. ∴ R = y ≥ -3 or [-3, ∞ )

7. y =|x| Domain: ℝ or (-∞, +∞) Range: y ≥0 or [0, +∞) Domain Range Your Turn

8. f(x) = (x + 2) 2 - 4 Domain: ℝ or (-∞, +∞) Range: y ≥ -4 or [-4, +∞) Domain Range Your Turn

9. Domain: x>-3 or (-3, +∞) Range: y>-4 or (-4, +∞) Domain Range, x ≠ -3 Your Turn

10. f(x) = -(x + 2) 3 -1 Domain: ℝ or (-∞, +∞) Range: ℝ or (-∞, +∞) Domain Range Your Turn

11. f(x) = 2 x+2 - 3 Domain: ℝ or (-∞, +∞) Range: y>-3 or (-3, +∞) Domain Range Your Turn

12. Determine the Domain and Range for Each Function From Their Graph D = -5 ≤ x ≤ 9 or [-5, 9] R = -5 ≤ y ≤ 5 or [-5, 5] D = -4 ≤ x<2 or [-4, 2) R = {-2, -1, 0} Your Turn

13. A relation is a correspondence that associates values of x with values of y. Relation x 2 + y 2 = 4 Example: The following equations define relations: The graph of a relation is the set of ordered pairs (x, y) for which the relation holds. y 2 = x x y (4, 2) (4, -2) x y (0, 2) (0, -2) y = x 2 x y

14. Functions - Vertical Line Test To determine whether or not a relation is in fact a function, we can draw a vertical line through the graph of the relation. If the vertical line intersects the graph more than once, then that means the graph of the relation is not a function. If the vertical line intersects the graph once then the graph shows that the relation is a function.

15. Examples: Vertical Line Test x y Function One point of intersection x y Not a Function Two points of intersection x y Vertical Line Test A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.

16. Vertical Line Test: Apply the vertical line test to determine which of the relations are functions. The graph does not pass the vertical line test. It is not a function. The graph passes the vertical line test. It is a function. x y x y Your Turn

17. Evaluating a Function Graphically Use the graph of g to evaluate g(–1). Example: Function g’s graph is shown.

18. Evaluating a Function Graphically (b) Find x = –1 on the x-axis. Move upward to the graph of g. (a)To evaluate g(-1) on the graph. (c) Move across to the y- axis. Read the y-value: g(–1) = 3. Solution

19. Your Turn Find f(2) f(2) = 3

20. Find f(3) Your Turn f(3) = -3

21. Find f(7), f(3), and f(1) Your Turn f(7) = 2 f(3) = 0 f(1) = DNE

22. Increasing and Decreasing Functions 1.Increasing function –The range values increase from left to right –The graph rises from left to right –Positive slope 2.Decreasing function –The range values decrease from left to right –The graph falls from left to right –Negative slope To decide whether a function is increasing, decreasing, or constant on an interval, ask yourself “What does the graph do as x goes from left to right?”

23. Graphs of Increasing and Decreasing Functions If could walk from left to right along the graph of an increasing function, it would be uphill. For a decreasing function, we would walk downhill.

24. Definition: Increasing, Decreasing, and Constant Functions Suppose that a function f is defined over an interval I. a.f increases on I if, whenever b.f decreases on I if, whenever c.f is constant on I if, for every Figure 7, pg. 2-4

25. Increasing and Decreasing Functions Increasing functions- rise from left to right Decreasing functions- falls from left to right Constant functions- flat (horizontal line) **A function that is increasing, decreasing, or constant, is described within its domain (in terms of x-values).** Ask yourself: What does y do as x goes from left to right?

26. Intervals of Increase or Decrease 5 x y -3 1 We need to identify where the function is increasing or decreasing Increasing: x<1 or (- , 1) Decreasing: x>1 or (1, +  )

27. Increasing, Decreasing, and Endpoints The concepts of increasing and decreasing apply only to intervals of the real number line and NOT to individual points. Do NOT say that the function f both increases and decreases at the point (0, 0). The point (0, 0) is the ‘turning point’. Decreasing: x<0 or (–∞, 0) Increasing: x>0 or (0, ∞)

28. Find the Intervals on the Domain in which the Function is Increasing, Decreasing, and/or Constant. Example Increasing: 3<x<5 Decreasing: x 5 Constant: -1<x<3

29. Your Turn Determine the intervals over which the function is increasing, decreasing, or constant. Solution: Ask “What is happening to the y-values as x is getting larger?”

30. Relative Minimum and Maximum Values of a Function A function f has a relative (local) maximum at x = c if there exists an open interval (r, s) containing c such that for all x between r and s. A function f has a relative (local) minimum at x = c if there exists an open interval (r, s) containing c such that for all x between r and s. Relative Maximums Relative Minimums

31. Assuming the graph is continuous (no break) at the point where the function changes from increasing to decreasing, that point is called a relative maximum point. Relative Maximum Values of a Function

32. In the same manner, a relative minimum point occurs when the graph changes from decreasing to increasing. Relative Minimum Values of a Function

33. **Maximum and Minimum values are based on y-values (or the output f(x)) of the function** Relative Maximum (Peak) –Highest value in some open interval Relative Minimum (Valley) –Lowest value in some open interval Relative Minimum and Maximum Values of a Function

34. Example: b. Relative Maximum e. Relative Maximum d. Relative Minimum s. Relative Minimum Indicate and label each maximum or minimum point on the function graph, from left to right.

35. Your Turn: Local Minimums b, c, & d Minimums r. Minimum e. Maximum s. Maximum Indicate and label each maximum or minimum point on the function graph, from left to right.

36. Your Turn: f(2)=1 Relative Minimum f(5)=2 Relative Minimum f(4)=4 Relative Maximum f(6)=3 Relative Maximum Find each maximum or minimum value and label as a maximum or minimum from left to right.

37. Example: –f(x) = -4x 2 – 7x + 3 –Answer: Max. (-0.875, 6.063) Example: –f(x) = -x 3 + x –Answer: Min. (-0.58, -0.38), Max. (0.58, 0.38) Your Turn: –F(x) = x 3 – 2x 2 –Answer: Max. (0, 0), Min. (1.3, -1.2) Use Graphing Calculator to Find Relative Minimum and Maximum Values of a Function

38. Graphing Step Functions and Piecewise – Defined Functions A step function is a specific kind of piecewise function in which the different parts of the equation look like steps when graphed.

39. Greatest Integer Function – Step Function The greatest integer function, denoted by ⟦ x ⟧ and defined as the largest integer less than or equal to x, has an infinite number of breaks or steps – one at each integer value in its domain. Also called the rounding-down or the floor function. Basic characteristics; –Domain: (-∞, +∞) –Range: {y: y = n, n є Z } –x-intercepts: in the interval [0, 1) –y-intercept: (0, 0) –Is a piecewise function. –Constant between each pair of consecutive integers. –Jumps vertically one unit at each integer value.

40. GREATEST INTEGER FUNCTION When greatest integer acts on a number, the value that represents the result is the greatest integer that is less than or equal to the given number. There are several descriptors in that expression. First of all you are looking only for an integer. Secondly, that integer must be less than or equal to the given number and finally, of all of the integers that satisfy the first two criteria, you want the greatest one. The brackets which indicate that this operation is to be performed is as shown: ‘[ ]’. Example: [1.97] = 1 There are many integers less than 1.97; {1, 0, -1, -2, -3, -4, …} Of all of them, ‘1’ is the greatest. Example: [-1.97] = -2 There are many integers less than -1.97; {-2, -3, -4, -5, -6, …} Of all of them, ‘-2’ is the greatest.

41. It may be helpful to visualize this function a little more clearly by using a number line. 012345678-2-3-4-5-6-7 Example: [6.31] = 6 6.31-6.31 Example: [-6.31] = -7 When you use this function, the answer is the integer on the immediate left on the number line. There is one exception. When the function acts on a number that is itself an integer. The answer is itself. Example: [5] = 5Example: [-5] = -5 Example: If there is an operation inside the greatest integer brackets, it must be performed before applying the function. Example: [5.5–3.6] = [1.9] = 1 Example: [3.6–5.5] = [-1.9] = -2 Example: [5.5+3.6] = [9.1] = 9 Example: [5.5  3.6] = [19.8] = 19

42. Calculating the Greatest Integer Function Your Turn 1)⟦ 33.4 ⟧ 2)⟦ 29/10 ⟧ 3)⟦ -5.2 ⟧ 4)⟦ 2 π⟧ 5)⟦ - √ 7 ⟧ 6)⟦ 16 ⟧ Answers 1)33 2)2 3)-6 4)6 5)-3 6)16 ⟦ x ⟧ is defined as the largest integer less than or equal to x. Example: ⟦ 2.5 ⟧ = 2

43. The greatest integer function can be used to construct a Cartesian graph. The simplest of which is demonstrated below. f(x) = [x] To see what the graph looks like, it is necessary to determine some ordered pairs which can be determined with a table of values. f(x) = [x]x f(0) = [0] = 00 f(1) = [1] = 11 f(2) = [2] = 22 f(3) = [3] = 33 f(-1) = [-1] = -1 f(-2) = [-2] = -2 -2 If we only choose integer values for x then we will not really see the function manifest itself. To do this we need to choose non- integer values. Graph of the Greatest Integer Function

44. f(x) = [x]xf(0) = [0] = 00 f(0.5) = [0.5] = 0 0.5 f(0.7) = [0.7] = 0 0.7 f(0.8) = [0.8] = 0 0.8 f(0.9) = [0.9] = 0 0.9 f(1) = [1] = 1 1 f(1.5) = [1.5] = 1 1.5 f(1.6) = [1.6] = 1 1.6 f(1.7) = [1.7] = 1 1.7 f(1.8) = [1.8] = 1 1.8 f(1.9) = [1.9] = 1 1.9 f(2) = [2] = 2 2 f(-1) = [-1] = -1 f(-0.5) =[-0.5]=-1 - 0.5f(-0.9) =[-0.9]=-1 - 0.9

45. When all these points are strung together the graph looks something like this – a series of steps. For this reason it is sometimes called the ‘STEP FUNCTION’. Notice that the left of each step begins with a closed (inclusive) point but the right of each step ends with an open (excluding point) We can’t really state the last (most right) x-value on each step because there is always another to the right of the last one you may name. So instead we describe the first x-value that is NOT on a given step. Example: (1,0)

46. Rather than place a long series of points on the graph, a line segment can be drawn for each step as shown to the right. The graphs shown thus far have been magnified to make a point. However, these graphs are usually shown at a normal scale as you can see on the next slide.

47. f(x) = [x] This is a rather tedious way to construct a graph and for this reason there is a more efficient way to construct it. Basically the greatest integer function can be presented with 4 parameters, as shown below. f(x) = a[bx - h] + k By observing the impact of these parameters, we can use them to predict the shape of the graph.

48. In these 3 examples, parameter ‘a’ is changed. As a increases, the distance between the steps increases. f(x) = [x] a = 1 f(x) = 2[x] a = 2 f(x) = 3[x] a = 3

49. When ‘a’ is negative, notice that the slope of the steps is changed. Downstairs instead of upstairs. But as ‘a’ changes from –1 to –2, the distance between steps increases. The further that ‘a’ is from 0, the greater the separation between steps. This can be described with a formula. Vertical distance between Steps = |a| f(x) = -[x] a = -1 f(x) = -2[x] a = -2

50. f(x) = [x] b = 1 f(x) = [2x] b = 2 As ‘b’ is increased from 1 to 2, each step gets shorter. Then as it is decreased to 0.5, the steps get longer.

51. When ‘b’ is negative, notice that the slope of the steps and the orientation of each step changes. It now is open on the left but closed on the right – opposite to the way it is when ‘b’ is positive. Notice that when both ‘a’ and ‘b’ are negative the slope of the steps becomes positive again. Both parameters affect the slope. Slope through closed points of each step = ab b > 0 b < 0 f(x) = [-x] b = -1 f(x) = -[-x] a = -1 b = -1 If ab > 0, steps are increasing. If ab < 0, steps are decreasing.

52. Parameters h and k do not have an impact in defining the shape of the graph. These parameters simply translate the graph. This translation can be taken into account by determining a starting point. The previous four formulae can be used to construct the greatest integer function provided that there is an ordered pair to start with – the starting point. For this we can use the y-intercept, f(0). Example: 4. Vertical distance between Steps = |a| =|2| = 2 5. Slope through closed points of each step = ab 2. Orientation of each step: b > 0 1. Starting point: (0,5)

53. 4. Vertical distance between Steps = |a| =|2| = 2 5. Slope through closed points of each step = ab 2. Orientation of each step: b > 0 1. Starting point: (0,5) After placing the first step, the closed point on the next step must be vertically aligned with the open point on the previous one, keeping in mind the slope of the steps (up or down). Example:

54. Your Turn: f(x) = 3[-x – 3] + 5 a = 3; b = -1; h = 3; k = 5 1. Starting point: (0,-4) f(0) = 3[-(0)-3]+5 =3(-3)+5 = -4 2. Orientation of each step: b < 0 5. Slope through closed points of each step = ab 4. Vertical distance between Steps =|3| = 3 = |a|

55. 5.Slope through closed points of each step = ab 1. Starting point: f(0) Five Steps to Construct Greatest Integer Graph b > 0 b < 0 2. Orientation of each step: 4. Vertical distance between Steps = |a| f(x) = a[bx - h] + k

56. Greatest Integer Function as a Piecewise Function

57. Graph of Greatest Integer Function as a Piecewise Function

58. Example: Application of a Step Function Downtown Parking charges a $5 base fee for parking through 1 hour, and $1 for each additional hour or fraction thereof. The maximum fee for 24 hours is $15. Sketch a graph of the function that describes this pricing scheme. Solution Sample of ordered pairs (hours, price): (.25,5), (.75,5), (1,5), (1.5,6), (1.75,6). During the 1 st hour: price = $5 During the 2 nd hour: price = $6 During the 3 rd hour: price = $7 During the 11 th hour: price = $15 It remains at $15 for the rest of the 24-hour period. Plot the graph on the interval (0,24].

59. The TI-89 command for the Greatest Integer function is floor (x). Y= CATALOG F floor( Graphing the Greatest Integer Function with a Graphing Calculator

60. Graph a Piecewise Function by Hand 1)Write all of the equations of the piecewise functions in y = mx + b form 2)Look at your first equation and its domain. Graph your first equation over that domain. 3)Put a closed circle on the beginning or ending points if that point is included in the domain (greater than or equal to, less than or equal to). If the point is not included in the domain, use an open circle (less than or greater than). 4)Repeat for the other equations of the function.

61. Example: Graph the piecewise function f(x). For all x’s < 1, use the top graph (to the left of 1) For all x’s ≥ 1, use the bottom graph (to the right of 1)

62. x=1 is the breaking point of the graph. To the left is the top equation. To the right is the bottom equation. Solution

63. Your Turn: Graph the piecewise function f(x).

64. Example: Graphing a Piecewise-Defined Function with a Graphing Calculator Graph each equation and its domain separately. –Example: –Input: y 1 =x+2|x<1 y 2 =(x-1) 2 |x ≥ 1 Your Turn: 1. 2.

65. Function Symmetry with Respect to the y-Axis Even Function If we were to “fold” the graph of f (x) = x 2 along the y-axis, the two halves would coincide exactly. We refer to this property as symmetry. Symmetry with Respect to the y-Axis – Even Function If a function f is defined so that for all x in its domain, then the graph of f is symmetric with respect to the y-axis.

66. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Even Functions have y-axis Symmetry

67. Function Symmetry with Respect to the Origin ODD Function If we “rotate” the graph of f (x) = x 3 180 ˚ about the origin, the two parts would coincide. We say that the graph is symmetric with respect to the origin. For example, Symmetry with Respect to the Origin - Odd Function If a function f is defined so that for all x in its domain, then the graph of f is symmetric with respect to the origin.

68. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Odd Functions have Origin Symmetry

69. Symmetry with Respect to the x-Axis If we “fold” the graph of x = y 2 along the x-axis, the two halves of the parabola coincide. This graph exhibits symmetry with respect to the x-axis. (Note, this relation is not a function. Use the vertical line test on its graph below.) e.g. Symmetry with Respect to the x-Axis If replacing y with –y in an equation results in the same equation, then the graph is symmetric with respect to the x-axis.

70. 2-7-6-5-4-3-21573 0468 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -7 We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. x-axis Symmetry

71. Even and Odd Functions To determine odd or even function, calculate f(-x). If f(-x) = f(x) even, if f(-x) = -f(x) odd, and if f(-x) ≠ f(x) & f(-x) ≠ -f(x) neither. Example Decide if the functions are even, odd, or neither. 1. 2. A function f is called an even function if for all x in the domain of f. (Its graph is symmetric with respect to the y-axis.) A function f is called an odd function if for all x in the domain of f. (Its graph is symmetric with respect to the origin.)

72. A function is even if f( -x) = f(x) for every number x in the domain. So if you substitute a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

73. A function is odd if f( -x) = - f(x) for every number x in the domain. So if you substitute a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

74. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

75. Your Turn 1)Determine whether f(x) = 5 – 3x is even, odd, or neither. –Answer: Neither 2)Determine whether f(x) = - x 2 + 3 is even, odd, or neither. –Answer : Even Functions may be described as even, odd, or neither. Even functions have graphs symmetric with the y-axis. Odd functions have graphs symmetric with the origin. To test algebraically whether a function is even or odd: Substitute -x for x in the function. The function is even if f(-x) = f(x). ( function is unchanged) The function is odd if f(-x) = -f(x). ( all signs change)

76. Homework Section 1.3, pg. 38 – 41: Vocabulary Check #1 – 6 all Exercises: #1-9 odd, 15-35 odd, 43, 47, 51, 53, 59 -69 odd, 95-100 all Read Section 1.4, pg. 42 – 47