C ollege A lgebra

1. College Algebra Functions and Graphs (Chapter1)

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

3. Functions 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. Chapter1

4. Functions 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. 4 Chapter1

5. Function Set B Set A y y y y y x x x x Domain Range Functions 5 Chapter1

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

7. Functions Definition of a Function 7 Chapter1

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

9. Functions 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) = x2 – 3x – 1. Find f (–2). f (x) = x2 – 3x – 1 f (–2) = 4 + 6 – 1Simplify. f (–2) = 9 The value of f at –2 is 9. 9 Chapter1

10. Functions 10 Chapter1

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

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

13. Functions 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 13 Chapter1

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

15. Functions Domain of a Function 15 Chapter1

16. Functions 16 Chapter1

17. Functions 17 Chapter1

18. y y y (4, 2) (0, 2) x x x (0, -2) (4, -2) Functions A relation is a correspondence that associates values of x with values of y. The graph of a relation is the set of ordered pairs (x, y) for which the relation holds. Example: The following equations define relations: y = x2 x2 + y2 = 4 y2= x 18 Chapter1

19. y y y x2 + y2 = 1 y = x2 y2= x x x x Functions 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 = x2is a function. Consider the graphs. 2 points of intersection 1 point of intersection 2 points of intersection 19 Chapter1

20. Functions Vertical Line Test:Apply the vertical line test to determine which of the relations are functions. y x = |y – 2| y x =2y – 1 x x The graph does not pass the vertical line test. It isnot a function. The graph passes the vertical line test. It is a function.

21. x y C D E 4 5 6 8 3 6 4 6 2 6 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.

22. domain range 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]

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. a. b. 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.

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. Range Domain 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, )

29. Range Domain 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, ).

30. Functions Example 8: Solution to Example 8:

31. Functions

32. Functions

33. Functions

34. Functions

35. Functions

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

38. Functions Example (1): 38 Chapter1

39. Functions Example (2): 39 Chapter1

40. Functions 40 Chapter1

41. Functions 41 Chapter1

42. Functions 42 Chapter1

43. Functions Example (3): 43 Chapter1

44. Functions 44 Chapter1

45. Functions Example (4): 45 Chapter1

46. Functions 46 Chapter1

47. Functions Example (5): 47 Chapter1

48. Functions 48 Chapter1

49. End of the Lecture Let Learning Continue