1. C ollege A lgebra Functions and Graphs (Chapter1) L:8 1 Instructor: Eng. Ahmed abo absa University of Palestine IT-College

2. 2 Objectives Chapter1 Cover the topics in Section ( 1-3):Functions After completing this section, you should be able to: 1.Know what a relation, function, domain and range are. 2.Find the domain and range of a relation. 3.Identify if a relation is a function or not. 4.Evaluate functional values.

3. 3 Functions Chapter1 Relation A relation is a set of ordered pairs where the first components of the ordered pairs are the input values and the second components are the output values. A relation is a rule of correspondence that relates two sets. For instance, the formula I = 500r describes a relation between the amount of interest I earned in one year and the interest rate r. In mathematics, relations are represented by sets of ordered pairs (x, y). Function A function is a relation that assigns to each input number EXACTLY ONE output number. Be careful. Not every relation is a function. A function has to fit the above definition to a tee.

4. 4 Functions Chapter1 Domain The domain is the set of all input values to which the rule applies. These are called your independent variables. These are the values that correspond to the first components of the ordered pairs it is associated with. Range The range is the set of all output values. These are called your dependent variables. These are the values that correspond to the second components of the ordered pairs it is associated with.

5. 5 Functions Chapter1 Function x y DomainRange Set A Set B

6. 6 Functions Chapter1 Example (1): Determine whether the relation represents y as a function of x. a) {(-2, 3), (0, 0), (2, 3), (4, -1)} b){(-1, 1), (-1, -1), (0, 3), (2, 4)} Function Not a Function

7. 7 Functions Chapter1 Definition of a Function

8. 8 Functions Chapter1 Input x The domain elements, x, can be thought of as the inputs and the range elements, f (x), can be thought of as the outputs. Output f (x) Function f

9. 9 Functions Chapter1 To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function. Example (1): Let f (x) = x 2 – 3x – 1. Find f (–2). f (x) = x 2 – 3x – 1 f (–2) = 4 + 6 – 1 Simplify. f (–2) = 9 The value of f at –2 is 9.

10. 10 Functions Chapter1

11. 11 Functions Chapter1 Example 3: Find and simplify using the function.

12. 12 Functions Chapter1 Constant Function A function of the form f(x) = C, where C is a constant. Example 4: Find the functional values h)0( and h(2) of the constant function h(x)=-5

13. 13 Functions Chapter1 Compound Function A compound function is also known as a piecewise function. The rule for specifying it is given by more than one expression. Example 5: Find the functional values f(1), f(3), and f(4) for the compound function

14. 14 Functions Chapter1 To find f(1),To find f(3),To find f(4),

15. 15 Functions Chapter1 Domain of a Function

16. 16 Functions Chapter1

17. 17 Functions Chapter1

18. 18 Functions Chapter1 x 2 + y 2 = 4 A relation is a correspondence that associates values of x with values of y. y 2 = x x y (4, 2) (4, -2) x y (0, 2) (0, -2) y = x 2 x y Example: The following equations define relations: The graph of a relation is the set of ordered pairs (x, y) for which the relation holds.

19. 19 Functions Chapter1 y 2 = x x y Vertical Line Test A relation is a function if no vertical line intersects its graph in more than one point. Of the relations y 2 = x, y = x 2, and x 2 + y 2 = 1 only y = x 2 is a function. Consider the graphs. x 2 + y 2 = 1 x y y = x 2 x y 2 points of intersection 1 point of intersection 2 points of intersection

20. Functions 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 = 2y – 1 x y x = | y – 2|

21. Functions Domain and Range In a relation, the set of all values of the independent variable (x) is the domain; the set of all values of the dependent variable (y) is the range. Example Give the domain and range of the relation. The domain is {4, 5, 6, 8}; the range is {C, D, E}. The mapping defines a function—each x-value corresponds to exactly one y-value. Give the domain and range of the relation. The domain is {3, 4, 2} and the range is {6}. The table defines a function because each different x-value corresponds to exactly one y-value. 45684568 CDECDE 62 64 63 yx

22. Functions Finding Domains and Ranges from Graphs Find the domain and range of the relation. The x-values of the points on the graph include all numbers between  3 and 3, inclusive. The y-values include all numbers between  2 and 2, inclusive. Domain = [  3, 3] Range = [  2, 2] range domain

23. Functions Definitions Agreement on Domain Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable. Vertical Line Test If each vertical line intersects a graph in at most one point, then the graph is that of a function.

24. Functions Example Use the vertical line test to determine whether each relation graphed is a function. The graph of (a) fails the vertical line test, it is not the graph of a function. Graph (b) represents a function. a. b.

25. Functions Identifying Function, Domains, and Ranges from Equations Decide whether each relation defines a function and give the domain and range. a) y = x + 5 b) c)

26. Functions Function Notation y = f(x) is called function notation We read f(x) as “f of x” (The parentheses do not indicate multiplication.) The letter f stand for function. f(x) is just another name for the dependent variable y. Example: We can write y = 10x + 3 as f(x) = 10x + 3

27. Functions Solutions a) y = x + 5 –Function: Each value of x corresponds to just one value of y so the relation defines a function. –Domain: Any real number of ( ,  ) –Range: Any real number ( ,  )

28. Functions Solutions continued b) –Domain: [1/3,  ) –Function: For any choice of x there is exactly one corresponding y-value. So the relation defines a function. –Range: y  0, or [0,  ) Range Domain

29. Functions Solutions continued c) –Function: There is exactly one value of y for each value in the domain, so this equation defines a function. –Domain: All real numbers except those that make the denominator 0. ( ,  1)  (  1,  ). –Range: Values of y can be positive or negative, but never 0. The range is the interval ( , 0)  (0,  ). Domain Range

30. Functions Example 8: Solution to Example 8 :

31. Functions

36. Variations of the Definition of Function A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. A function is a set of ordered pairs in which no first component is repeated. A function is a rule or correspondence that assigns exactly one range value to each domain value.

37. 37 Chapter1 Examples on: -Function Graph. -Domain and Range of Functions

38. 38 Functions Chapter1 Example (1):

39. 39 Functions Chapter1 Example (2):

40. 40 Functions Chapter1

41. 41 Functions Chapter1

42. 42 Functions Chapter1

43. 43 Functions Chapter1 Example (3):

44. 44 Functions Chapter1

45. 45 Functions Chapter1 Example (4):

46. 46 Functions Chapter1

47. 47 Functions Chapter1 Example (5):

48. 48 Functions Chapter1