2. Represent Relations and Functions Objectives: 1.To find the domain and range of a relation or function 2.To determine if a relation is a function 3.To classify and evaluate functions 4.To distinguish between discrete and continuous functions

3. Vocabulary RelationFunction InputOutput DomainRange Independent Variable Dependent Variable

4. Objective 1 You will be able to find the domain and range of a relation (or function)

5. Relation relation A mathematical relation is the pairing up (mapping) of inputs and outputs. What’s the domain and range of each relation?

6. Relations relation A mathematical relation is the pairing up (mapping) of inputs and outputs. Range: The set of all output values Range: The set of all output values Re lat ion Domain: The set of all input values Domain: The set of all input values

7. Exercise 1 Consider the relation given by the ordered pairs (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3). 1.Identify the domain and range

8. Exercise 1 Consider the relation given by the ordered pairs (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3). 2.Represent the relation as a graph and as a mapping diagram

9. Objective 2 You will be able to tell if a relation is a function

10. Calvin and Hobbes! function A toaster is an example of a function. You put in bread, the toaster performs a toasting function, and out pops toasted bread.

11. Calvin and Hobbes! What comes out of a toaster? It depends on what you put in. You can’t input bread and expect a waffle!

12. What’s Your Function? A function is a dependent relation Output depends on the input Relations Functions function A function is a relation in which each input has exactly one output.

13. What’s Your Function? Each output does not necessarily have only one input Relations Functions function A function is a relation in which each input has exactly one output.

14. BIG If you think of the input as a boy and the output as a girl, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIG trouble.

15. Functional Relation Non-Functional Relation

16. Another Functional Relation

17. What’s a Function Look Like?

21. Exercise 2a Tell whether or not each table represents a function. Give the domain and range of each relationship.

22. Exercise 2b cardinality The size of a set is called its cardinality. What must be true about the cardinalities of the domain and range of any function?

23. Exercise 3 Which sets of ordered pairs represent functions? 1.{(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)} 2.{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} 3.{(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)} 4.{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

24. Exercise 4 Which of the following graphs represent functions? What is an easy way to tell that each input has only one output?

25. Vertical Line Test A relation is a function iff no vertical line intersects the graph of the relation at more than one point Function Not a Function If it does, then an input has more than one output.

26. Objective 1 You will be able to find the domain and range of a relation (or function) -Revisited-

27. Domain and Range: Graphs Domain: All real numbers

28. Domain and Range: Graphs Range: Greater than or equal to -4

29. Exercise 5 Determine the domain and range of each function.

30. Domain and Range: Equations

31. Exercise 6 Determine the domain of each function.

32. Protip: Domains of Equations When you have to find the domain of a function given its equation there’s really only two limiting factors: The denominator of any fractions can’t be zero Square roots can’t be negative

33. Objective 3 You will be able to classify and evaluate functions

34. Dependent Quantities dependent depends Functions can also be thought of as dependent relations. In a function, the value of the output depends on the value of the input. Independent Quantity Input values Domain Dependent Quantity Output values Range

35. Exercise 7 The number of pretzels, p, that can be packaged in a box with a volume of V cubic units is given by the equation p = 45 V + 10. In this relationship, which is the dependent variable?

36. Function Notation

37. Exercise 8e

38. Flavors of Functions Functions come in a variety of flavors. You will need to be able to distinguish a linear from a nonlinear function. Linear FunctionNonlinear Function f ( x ) = 3 xf ( x ) = x 2 – 2 x + 5 g ( x ) = ½ x – 5 g ( x ) = 1/ x h ( x ) = 15 – 5 xh ( x ) = | x | + 2

39. Exercise 8c Classify each of the following functions as linear or nonlinear.

40. Exercise 9

41. Objective 4 You will be able to tell the difference between continuous and discrete functions

42. Analog vs. Digital Analog: A signal created by some physical process Digital: A numerical representation of an analog signal created by samples  Sound, temperature, etc.  Contains an infinite amount of data  Not continuous = set of points  Contains a finite amount of data

43. Digital Signal Processing Original Analog Signal Digital Samples

44. Digital Signal Processing Digital signal processing is about converting an analog signal into digital information, doing something to it, and usually converting it back into an analog signal.

45. Continuous vs. Discrete Continuous Function: A function whose graph consists of an unbroken curve Discrete Function: A function whose graph consists of a set of discontinuous points

46. Exercise 10 Determine whether each situation describes a continuous or a discrete function. Then state a realistic domain. 1.The cheerleaders are selling candy bars for $1 each to pay for new pom-poms. The function f ( x ) gives the amount of money collected after selling x bars.

47. Exercise 10 Determine whether each situation describes a continuous or a discrete function. Then state a realistic domain. 2.Kenny determined that his shower head releases 1.9 gallons of water per minute. The function V ( x ) gives the volume of water released after x minutes.

48. Represent Relations and Functions Objectives: 1.To find the domain and range of a relation or function 2.To determine if a relation is a function 3.To classify and evaluate functions 4.To distinguish between discrete and continuous functions