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# Functions

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teaching the basic of functions in mathematics.

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### Functions

1. 1. Functions Students will determine if a given equation is a function using the vertical line test and evaluate functions given member(s) of the domain. Compiled by : Motlalepula Mokhele Student at the University of Johannesburg (2014)
2. 2. 1  5 2  6 3  7 4  8 5  9 The Rule is ‘ADD 4’
3. 3. Ahmed  Paris Peter  London Ali Dubai Jaweria  New York Cyprus Hamad  Has Visited There are MANY arrows from each person and each place is related to MANY People. It is a MANY to MANY relation.
4. 4. Person Bilal Has A Mass of  Kg 62 Peter  Salma  Alaa  George  Aziz 64 66  In this case each person has only one mass, yet several people have the same Mass. This is a MANY to ONE relationship
5. 5. Is the length of cm  14  object Pen Pencil Ruler  30  Needle Stick Here one amount is the length of many objects. This is a ONE to MANY relationship
6. 6. FUNCTIONS  Many to One Relationship  One to One Relationship
7. 7. x2x+1 A B 0 1 2 3 4 Domain 1 2 3 4 5 6 7 8 9 Co-domain Image Set (Range)
8. 8. f : x  x2 + 4 f( x) = x2 + 4 The upper function is read as follows:‘Function f such that x is mapped onto x2+4
9. 9. Lets look at some function Type questions If f ( x ) = x 2 + 4 and g ( x ) = 1 − x 2 F ind f ( 2 ) F ind g ( 3 ) 2 f(x) = x 2 + 4 2 =8 g(x) = 1 − x 2 3 3 = -8
10. 10. Consider the function f(x) = 3x − 1 x We can consider this as two simpler functions illustrated as a flow diagram 3x Multiply by 3 Subtract 1 3x − 1 Consider the function f : x  (2x + 5) 2 x Multiply by 2 2x Add 5 2x + 5 Square (2x + 5) 2
11. 11. Consider 2 functions f : x  3x + 2 and g(x) : x  x 2 fg is a composite function, where g is performed first and then f is performed on the result of g. The function fg may be found using a flow diagram x square g Thus fg = 3x 2 + 2 x2 Multiply by 3 3x 2 f Add 2 3x 2 + 2
12. 12. x2 3x + 2 f g 2 4 2 fg (x) 3x 2 + 2 14
13. 13. Consider the function f(x) = 5x − 2 3 Here is its flow diagram 5 x -2 5x x Multiply by 5 Subtract 2 f(x) = 5x − 2 3 Divide by three Draw a new flow diagram in reverse!. Start from the right and go left… 3 x +2 5 3x 3 x +2 Divide by 5 And so f −1 (x) = 3x + 2 5 Add two x Multiply by three
14. 14. (b) (a) (c) (d) (a) and (c)
15. 15. (b) (a) (c) (d) (a) and (c)
16. 16. Functions and Their Graphs
17. 17. Definition of Relation  Relation – a set of ordered pairs, which contains the pairs of abscissa and ordinate. The first number in each ordered pair is the x-value or the abscissa, and the second number in each ordered pair is the y-value, or the ordinate.    Domain is the set of all the abscissas, and range is the set of all ordinates.
18. 18. Relations  A relation may also be shown using a table of values or through the use of a mapping diagram.  Illustration:  Using a table. Domain 0 1 2 3 4 7 Using a mapping diagram. Range 1 2 3 4 5 8
19. 19. Definition of Function  Function – a characteristic of set of values where each element of the domain has only one that corresponds with it in the range. It is denoted by any letter of the English alphabet.  The function notation f(x) means the value of function f using the independent number x.
20. 20. Example 1a.  Given the ordered pairs below, determine if it is a mere relation or a function.  (0,1) , (1, 2), (2, 3), (3, 4), (4, 5), (7, 8)  Answer:  For every given x-value there is a corresponding unique y-value. Therefore, the relation is a function.
21. 21. Example 1b. Which relation represents a function? A. {(1,3), (2, 4), (3,5), (5, 1)} B. {(1, 0), (0,1), (1, -1)} C. {(2, 3), (3, 2), (4, 5), (3, 7)} D. {(0, 0), (0, 2)}  Answer: A 
22. 22. Example 1c.  Which mapping diagram does not represent a function?  A. B.  C. D.
23. 23. Types of Functions
24. 24. 1.Increasing, Decreasing, and Constant Functions A function is increasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) < f (x2). A function is decreasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) > f (x2). A function is constant on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) = f (x2). (x2, f (x2)) (x1, f (x1)) Increasing f (x1) < f (x2) (x1, f (x1)) (x2, f (x2)) (x1, f (x1)) (x2, f (x2)) Decreasing f (x1) < f (x2) Constant f (x1) < f (x2)
25. 25. Example 8a. Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. a b 5 . 4 5 . 3 4 3 2 1 1 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 -5 -4 -3 -2 -1 -1 -2 -3 2 3 4 5 -3 -4 -5 1 -4 -5 Solutio a. n The function is decreasing on the interval (-∞, 0), increasing on the interval (0, 2), and decreasing on the interval (2, ∞).
26. 26. Example 8a. Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. a b 5 . 4 5 . 3 4 3 2 1 1 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 -5 -4 -3 -2 -1 -1 -2 -3 2 3 4 5 -3 -4 -5 1 -4 -5 Solution: b. • Although the function's equations are not given, the graph indicates that the function is defined in two pieces. • The part of the graph to the left of the y-axis shows that the function is constant on the interval (-∞, 0). • The part to the right of the y-axis shows that the function is increasing on the interval [0,∞).
27. 27. Example 8b. Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. Decreasing on (-∞, 0); Increasing on (0, ∞)
28. 28. Example 8b. Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. Increasing on (-∞, 2); Constant on (2, ∞)
29. 29. Example 8c. Describe the increasing, decreasing, or constant behavior of each function whose graph is shown. Increasing on (-∞,∞)
30. 30. 2.Continuous and Discontinuous Functions A continuous function is represented by a graph which may be drawn using a continuous line or curve, while a discontinuous function is represented by a graph which has some gaps, holes or breaks (discontinuities).
31. 31. 3.Periodic Functions A periodic function is a function whose values repeat in periods or regular intervals. y = tan(x) y = cos(x)
32. 32. A linear function is a function of the form f(x) = mx +b where m and b are real numbers and m ≠ 0. Domain: the set of real numbers Range: the set of real numbers Graph: straight line Example: f(x) = 2 - x
33. 33. 5. Quadratic Functions A quadratic function is a function of the form f(x) = ax2 +bx +c where a, b and c are real numbers and a ≠ 0. Domain: the set of real numbers Graph: parabola Examples: parabolas parabolas opening upward opening downward
34. 34. Graphs of Quadratic Functions The graph of any quadratic function is called a parabola. Parabolas are shaped like cups, as shown in the graph below. If the coefficient of x2 is positive, the parabola opens upward; otherwise, the parabola opens downward. The vertex (or turning point) is the minimum or maximum point.
35. 35. Evaluation of Functions
36. 36. Example 2. If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3) c. f (-x) Solution a. We find f (2) by substituting 2 for x in the equation. f (2) = 22 + 3 • 2 + 5 = 4 + 6 + 5 = 15 Thus, f (2) = 15.
37. 37. Example 2. If f (x) = x2 + 3x + 5, evaluate: b. f (x + 3) Solution b. We find f (x + 3) by substituting (x + 3) for x in the equation. f (x + 3) = (x + 3)2 + 3(x + 3) + 5 Equivalently, f (x + 3) = (x + 3)2 + 3(x + 3) + 5 = x2 + 6x + 9 + 3x + 9 + 5 = x2 + 9x + 23.
38. 38. Example 2. If f (x) = x2 + 3x + 5, evaluate: c. f (-x) Solution c. We find f (-x) by substituting (-x) for x in the equation. f (-x) = (-x)2 + 3(-x) + 5 Equivalently, f (-x) = (-x)2 + 3(-x) + 5 = x2 –3x + 5.
39. 39. Example 3a.  Which is the range of the relation described by y = 3x – 8 if its domain is {-1, 0, 1}?  A) {-11, 8, 5}  B) {-5, 0 5}  C) {-11, -8, -5}  D) {0, 3, 5}
40. 40. Operations on Functions
41. 41. Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum, the difference, the product , and the quotient are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows:  Sum: (f + g)(x) = f (x)+g(x)  Difference: (f – g)(x) = f (x) – g(x)  Product: (f • g)(x) = f (x) • g(x)  Quotient: (f / g)(x) = f (x)/g(x), g(x) ≠ 0
42. 42. Example 4a. Let f(x) = 2x+1 and g(x) = x2 - 2. Find a. (f + g) (x) c.(g – f) (x) e. (f / g) (x) b. (f – g) (x) d. (f ∙ g) (x) f. (g/f) (x) Solution: a. (f + g) (x) = f(x) + g( x) = (2x+1 )+ (x2 – 2) = x2 + 2x - 1 b. (f – g)(x) = f(x) - g(x) = (2x+1) - (x 2 - 2) = -x2 + 2x + 3 c. (g – f)(x) = g(x) - f(x) = (x2 - 2) – (2x +1) = x2 - 2x - 3 d. (f ∙ g)(x) = f(x) ∙ g(x) = (2x+1)(x 2 - 2) = 2x3 + x2 - 4x - 2 e. (f/g)(x) = f(x)/g(x) = (2x+1)/(x 2 - 2), x≠± 2 2 f. (g/f)(x) = g(x)/f(x) = (x - 2)/(2x +1), 1 x≠− 2
43. 43. Example 5a. Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f ○ g)(x) b. (g ○ f)(x) Solution a. We begin with (f o g)(x), the composition of f with g. Because (f o g)(x) means f (g(x)), we must replace each occurrence of x in the equation for f by g(x). f (x) = 3x – 4 (f ○ g)(x) = f (g(x)) = 3(g(x)) – 4 = 3(x2 + 6) – 4 = 3x2 + 18 – 4 = 3x2 + 14 Thus, (f ○ g)(x) = 3x2 + 14.
44. 44. Example 5a. Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f ○ g)(x) b. (g ○ f)(x) Solution b. Next, we find (g o f )(x), the composition of g with f. Because (g o f )(x) means g(f (x)), we must replace each occurrence of x in the equation for g by f (x). g(x) = x2 + 6 (g ○ f )(x) = g(f (x)) = (f (x))2 + 6 = (3x – 4)2 + 6 = 9x2 – 24x + 16 + 6 = 9x2 – 24x + 22 Notice that (f ○ g)(x) is not the same as (g ○ f )(x).
45. 45. Graphs of Relations and Functions
46. 46. Graph of a Function  If f is a function, then the graph of f is the set of all points (x,y) in the Cartesian plane for which (x,y) is an ordered pair in f.  The graph of a function can be intersected by a vertical line in at most one point.  Vertical Line Test  If a vertical line intersects a graph more than once, then the graph is not the graph of a function.
47. 47. Example 6a.  Determine if the graph is a graph of a function or just a graph of a relation. 8 6 4 2 5 -2 -4 10 15 graph of a relation
48. 48. Example 6b.  Determine if the graph is a graph of a function or just a graph of a relation. graph of a function
49. 49. Example 6c.  Determine if the graph is a graph of a function or just a graph of a relation. graph of a relation
50. 50. Example 6d.  Determine if the graph is a graph of a function or just a graph of a relation. 16 14 12 10 8 6 4 graph of a relation 2 A 15 10 5 5 2 4 6 8 10 15 20 25
51. 51. Example 6e.  Determine if the graph is a graph of a function or just a graph of a relation. 4 3 2 1 -6 -4 -2 2 -1 -2 -3 -4 4 6 graph of a relation
52. 52. Example 6f.  Determine if the graph is a graph of a function or just a graph of a relation. 6 4 2 -10 -5 5 -2 -4 -6 10 graph of a relation
53. 53. Example 6g.  Determine if the graph is a graph of a function or just a graph of a relation. 3 1 -3 -2 -1 -1 -2 -3 -5 1 2 3 4 graph of a function
54. 54. Graphing Parabolas  Given 4. Find any x-intercepts by replacing f (x) with 0. Solve the resulting quadratic equation for x. 5. Find the y-intercept by replacing x with zero. 6. Plot the intercepts and vertex. Connect these points with a smooth curve that is shaped like a cup. f(x) = ax2 + bx +c
55. 55. Graphing Parabolas  Given 1. Determine whether the parabola opens upward or downward. If a > 0, it opens upward. If a < 0, it opens downward. 2. Determine the vertex of the parabola. The vertex is f(x) = ax2 + bx +c  − b 4ac − b 2  ,   4a   2a  The axis of symmetry is −b x= 2a The axis of symmetry divides the parabola into two equal parts such that one part is a mirror image of the other.
56. 56. This powerpoint was kindly donated to www.worldofteaching.com http://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching. http://rechneronline.de/function-graphs/ http://www.coolmath.com/graphit/ http://www.slideshare.net/bloodyheartjinxz/presentation1-26579533?v=