1. Definition of a Relation relation domain range A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation. Domain Range Mrs. Rivas International Studies Charter School.

2. Mrs. Rivas International Studies Charter School. Finding the Domain and Range of a Relation Example 1 Find the domain and range of the relation: {(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (40, 21.2)}. Domain: Domain: {0, 10, 20, 30, 40} Range: Range: {9.1, 6.7, 10.7, 13.2, 21.2}

3. Mrs. Rivas International Studies Charter School. Definition of a Function function domainrange exactly one A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. 12341234 124124 Domain Range

4. Mrs. Rivas International Studies Charter School. Example 2 Determining Whether a Relation is a Function Determine whether the relation is a function: {(1, 6), (2, 6), (3, 8), (4, 9)} A the relation IS a function. It shows that every element in the domain corresponds to exactly one element of the range. Thus, the relation IS a function. B {(6, 1), (6, 2), (8, 3), (9, 4)} the relation is NOT a function. It shows that one element in the domain corresponds to more than one element of the range. Thus, the relation is NOT a function.

5. Mrs. Rivas International Studies Charter School. Functions as Equations For each value of x there is only one value of y.

6. Mrs. Rivas International Studies Charter School. Functions as Equations This graph doesn’t define y as a function of x.

7. Mrs. Rivas International Studies Charter School. Example 3 Determining Whether an Equation Represents a Function A B

8. Mrs. Rivas International Studies Charter School. Example 3 Determining Whether an Equation Represents a Function B

9. Mrs. Rivas International Studies Charter School. The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. Use the vertical line test to identify graphs in which y is a function of x. functionnot a functionfunction not a function

10. Mrs. Rivas International Studies Charter School. Functions as Equations This graph doesn’t define y as a function of x.

11. Mrs. Rivas International Studies Charter School. Function Notation

12. Mrs. Rivas International Studies Charter School. Evaluating a Function Example 4 A

13. Mrs. Rivas International Studies Charter School. Evaluating a Function Example 4 B

14. Mrs. Rivas International Studies Charter School. Evaluating a Function Example 4 C

15. Mrs. Rivas International Studies Charter School. Identifying Domain and Range from a Function’s Graph find the domain To find the domain of a function from it’s graph, look for all the inputs on the x-axis that correspond to points on the graph. find the range To find the range of a function from it’s graph, look for all the outputs on the y-axis that correspond to points on the graph.

16. Mrs. Rivas International Studies Charter School. Using Set-Builder Notation

17. Mrs. Rivas International Studies Charter School. Using Interval Notation

18. Mrs. Rivas International Studies Charter School.

19. Mrs. Rivas International Studies Charter School.

20. Mrs. Rivas International Studies Charter School. Identifying the Domain and Range of a Function from Its Graph Example 5 Use the graph of the function to identify its domain and its range. Domain Range

21. Mrs. Rivas International Studies Charter School. Identifying the Domain and Range of a Function from Its Graph Example 5 Use the graph of the function to identify its domain and its range. Domain Range

22. Mrs. Rivas International Studies Charter School. Identifying Intercepts from a Function’s Graph To find the x-intercepts, look for the points at which the graph crosses the x-axis. To find the y-intercept, look for the point at which the graph crosses the y-axis. A function can have more than one x-intercept but at most one y-intercept.

23. Mrs. Rivas International Studies Charter School. Identifying Intercepts from a Function’s Graph Example 6 Identify the x- and y-intercepts for the graph of f(x). The x-intercepts are ( – 3, 0), ( – 1, 0) and (2, 0). The y-intercept is (0, –6).

24. Mrs. Rivas International Studies Charter School. Increasing, Decreasing, and Constant Functions

25. Mrs. Rivas International Studies Charter School. Increasing, Decreasing, and Constant Functions

26. Mrs. Rivas International Studies Charter School. Increasing, Decreasing, and Constant Functions

27. Mrs. Rivas International Studies Charter School. Intervals on Which a Function Increases, Decreases, or is Constant Example 7 State the intervals on which the given function is increasing, decreasing, or constant. increasing on decreasing on increasing on

28. Minimum and maximum value Mrs. Rivas International Studies Charter School. Definitions of Relative Maximum and Relative Minimum

29. Mrs. Rivas International Studies Charter School. Definitions of Relative Maximum and Relative Minimum

30. Mrs. Rivas International Studies Charter School. Definitions of Relative Maximum and Relative Minimum

31. Mrs. Rivas International Studies Charter School. relative minimum f has a relative minimum at (1,-1)

32. Domain and Range Mrs. Rivas International Studies Charter School.

33. What is the is the domain and range of the following graphs. A)B) Domain (h) = (-∞, ∞). Range (k) = [-3, ∞). Domain (h) = (-∞, ∞). Range (k) = (-∞, 2]. Mrs. Rivas International Studies Charter School.

34. Mrs. Rivas International Studies Charter School. Definitions of Even and Odd Functions

35. Mrs. Rivas International Studies Charter School. Identifying Even or Odd Functions Example 8 Determine whether the function is even, odd, or neither. The function is not even. The function is not odd. is neither The function is neither odd nor even.

36. Mrs. Rivas International Studies Charter School. Even Functions and y-Axis Symmetry Odd Functions and Origin Symmetry

37. Mrs. Rivas International Studies Charter School. Example 9 odd This linear graph goes through the origin. If I rotate the graph 180° around the origin, I'll get the same picture. So this graph is odd. (The function would not be odd if this line didn't go through the origin.) even This parabola's vertex is on the y- axis, so the axis of symmetry is the y- axis. That means that the function is even. This cubic is centered on the origin. If I rotate the graph 180° around the origin, I'll get the same picture. So this graph is odd. Determine from the graphs whether the displayed functions are even, odd, or neither.

38. Mrs. Rivas International Studies Charter School. Example 9 Determine from the graphs whether the displayed functions are even, odd, or neither. neither This cubic is centered at the point (0, –3). This graph is symmetric, but not about the origin or the y-axis. So this function is neither even nor odd. neither This square root has no symmetry. The function is neither even nor odd. odd This cube root is centered on the origin, so this function is odd.

39. Mrs. Rivas International Studies Charter School. Example 9 Determine from the graphs whether the displayed functions are even, odd, or neither. even This graph looks like a bell-shaped curve. Since it is mirrored around the y-axis, the function is even.

40. Mrs. Rivas International Studies Charter School. Piecewise Functions A function that is defined by two (or more) equations over a specified domain is called a piecewise function.

41. Mrs. Rivas International Studies Charter School. Evaluating a Piecewise Function Example 10 Given the function Find C(40) Find C(80)

42. Mrs. Rivas International Studies Charter School. Graphing a Piecewise Function Example 11 Graph the piecewise function defined by We will graph f in two parts, using a partial table of coordinates for each piece of the graph.

43. Mrs. Rivas International Studies Charter School. Graphing a Piecewise Function Example 11 The graph of

44. Mrs. Rivas International Studies Charter School. Definition of the Difference Quotient of a Function

45. Mrs. Rivas International Studies Charter School. Definition of the Difference Quotient of a Function If find and simplify the expression.

46. Mrs. Rivas International Studies Charter School. Definition of the Difference Quotient of a Function If find and simplify the expression.

47. Mrs. Rivas International Studies Charter School 1-2 Pages: 1-2 Pages: 159-161 # 1, 3, 11, 15, 16, 20, 31, 35, 40, 43, 45, 47, 49, 51, 78, 80, 82, 84, 90 1-3 Pages: 1-3 Pages: 172-176 # 1, 3, 10, 12, 13, 18, 20, 22, 33, 36, 38, 40, 43, 46, 48, 51, 55, 58, 60, 64, 76