1. SWBAT… define and evaluate functions Agenda 1. Warm-Up (5 min) 2. Quiz – piecewise functions (6 min) 3. Notes on functions (25 min) 4. OYO problems (10 min) Warm-Up : Does the table below show a linear relationship? HW#1: Evaluating functions (page 1) Wed, 10/19 xy 0.50.0030 1.00.0066 1.50.0102 2.00.0138 Answer: Yes, the rate of change is constant. Complete this problem on graph paper – to be turned in!

2. Graph the piecewise function (on graph paper)

3. Ms. Sophia Papaefthimiou Infinity HS

4. Objectives Today: 1. To define a function 2. To learn function notation 3. To evaluate functions Tomorrow: 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function 4. To write a function as an ordered pair

5. What is a function? A function is like a machine: it has an input and an output. And the output is related somehow to the input.

6. Function Notation The most common name is "f", but you can have other names like "g" What goes into the function is put inside parentheses after the name of the function Example: f(x) (pronounced “f of x”) shows you the function is called "f", and "x" goes in. Question: What if a function was called “g” and “a” went into it? How would you write the function?

7. Output Value Range Dependent Variable These are all equivalent names for the y. Input Value Domain Independent Variable These are all equivalent names for the x. Name of the function Function Notation: The Symbolic Form

8. Function Notation Function notation replaces the y in an equation with f(x) Example: Given y = 3x + 2, write the equation in function notation y = 3x + 2 f(x) = 3x + 2 Question: Given y = x 2,write the equation in function notation.

9. Function Notation You used to say "y = 2x + 3; find the value of y when x = -1“ y = 2x + 3; find the value of y when x = -1 y = 2(-1) + 3 y = -2 + 3 y = 1 Now you say "f(x) = 2x + 3; find f(-1)" (pronounced as "f of x is two x plus three; find f of negative one") You do exactly the same thing in either case: you plug in -1 for x, multiply by 2, and then add the 3, to get a value of 1. 1 f(-1) = 2x + 3

10. Ex: If f(x) = 2x + 3 find: 1. f(-9) 2. f(4x) 3. f(-1) + f(-9) 4. f(t – 5)

11. Warm-Up: Directions: If f(x) = 2x – 4 and g(x) = x² – 4x, find each value: 1. f(-3) 2. f(3x) 3. g(t) 4. f(q + 1) 5. f(2) + g(-2) 6. f(g(-2)) (Hint: Start from the inside out. Find g(-2) first)

12. Revisit our objectives Today: 1. To define a function 2. To learn function notation 3. To evaluate functions

13. SWBAT… list the domain and range of functions Agenda 1. Warm-Up (5 min) 2. Review hw#8 – page 1 (10 min) 3. Notes on functions (30 min) Warm-Up: 1.) Take out hw#8: Functions 2.) Open your note books from Wednesday – we are going to continue our notes HW#8: Functions (page 2 – 4) Thurs, 10/20

14. Objectives Wednesday 1. To define a function 2. To learn function notation 3. To evaluate functions Today 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function 4. To write a function as an ordered pair

15. Function Mapping A set of points or equation where every input has exactly one output. For each x-value there is only one y-value In other words, x can not be repeated

16. This is a function! There is only one arrow coming from each x. There is only one y for each x. In other words, x can not be repeated This is a function! There is only one arrow coming from each x There is only one y for each x. It just so happens that it's always the same y for each x. Function Mapping (cont’d)

17. This one is not a function. There are two arrows coming from the number 1. The number 1 is associated with two different range elements. In order words, x is repeated. Function Mapping (cont’d)

18. Vertical Line Test No mater where we drop a vertical line, if the vertical line only hits the graph once, it is a function. So, this graph is a function! Draw a graph, that would NOT pass the vertical line test.

19. Vertical Line Test (cont’d) Intersect at two points These graphs are not functions

20. SWBAT… find the domain and range of functions Agenda 1. WU (10 min) 2. Domain and range (10 min) 3. 10 practice problems (20 min) Warm-Up: For problems 1 - 2: a. State the domain. b. State the range. c. Is this a function? Explain. 1. {(3,3), (4,3), (2,1), (6,5)} 2. {(2,1), (5,6), (2,3), (6,7) } HW#1: Functions – Page 3 & 4 Mon, 10/24

21. Domain and Range Domain: What can go into a function. The set of all x values in a function. Range: What comes out of a function. The set of all y values in a function.

22. Domain and Range (cont’d) f(x) = x 2 – 2 The domain is the set of all real numbers. The range is y ≥ -2.

23. Question 1/10 Question: For the function f(x) = x 2, if the domain is {1, 2, 3}, what is the range?

24. Question 2/10 Given f(x) = 3x – 5 and the domain is {0, 2, -1} find the range

25. Question 3/10 If f(x) = -x 2 find a. f(3) b. f(-3)

26. Question 4/10 Function f is defined by f(x) = -2x 2 + 6x – 3 1. Find f(-2) 2. Write as an ordered pair

27. Question 5/10 Does the diagram represent a function? Why? 24 3 15 017017

28. Question 6/10 Is the graph shown below that of a function? Why?

29. Question 7/10 Suppose h(w) = 2w. What is h(v)? Answer: h(v) = 2(v) h(v) = 2v

30. Question 8/10 What does the function notation g(7) represent? (what is the input and output) Answer: g(7) is the output, the input is 7

31. Question 9/10 Suppose g(x) = 3x + 2. Describe, in words, what the function g does. Answer: The function g takes an input, multiplies by 3, and then adds 2.

32. Question 10/10 Write in function notation “the function g takes an input y adds 3, and then multiplies by 2.” Answer: g(y) = 2(y + 3)

33. HW#1: Functions Now you should be able to finish Pages 1 – 4 of the functions hw (collected tomorrow) Use your notes/practice problems as you are finishing the hw

34. SWBAT… find the domain and range of functions Agenda 1. WU (15 min) 2. Domain & range practice problems (20 min) Warm-Up: Do the problems on the back of this week’s agenda (read the examples on the top of the sheet) HW#1: Functions – Page 5 – 6 Tues, 10/25

35. Real Numbers: All numbers on the number line. This includes positives and negatives, integers and rational numbers, square roots, cube roots, π, etc.

36. The domain of a function is the set of numbers that you can plug into the function and get out something that makes sense. When finding the domain, remember: 1. The denominator (bottom) of a fraction cannot be zero 2. The values under a square root sign must be positive Ex 1: What is the domain of ?

38. The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function. When finding the domain, remember: 1. Substitute different x-values into the expression to see what is happening for y. 2. Draw a sketch! In math, it's very true that a picture is worth a thousand words Ex: What is the range of ?

40. Ex 2: What is the domain of ? Domain: x ≥ 1 Range: y ≥ 0

41. Ex 2: Find the domain and range for the function f(x) = x 2 + 2. Explain. The domain is “all real numbers of x” because there are no restrictions on the value of x. The range is “y ≥ 2” since x 2 + 2 is never less than 2.

42. Ex 3: Find the domain and range for the function f(x) = x 2 – 5. Explain. The domain is “all real numbers of x” because there are no restrictions on the value of x. The range is “y ≥ -5” since x 2 – 5 is never less than -5.

43. To use the symbols of algebra, we could write the domain as Does that look like a foreign language? Let’s translate:

44. The curly braces just tell us we have a set of numbers.

45. The x reminds us that our set contains x- values.

46. The colon says, such that

47. The symbol that looks like an e says, belongs to...

48. And the cursive, or script, R is short for the set of real numbers.

49. R, the set of real numbers.” So we read it, “The set of xsuch that x belongs to

50. Question What is the domain of ? “The set of x such that x does not equal 0.”

51. What is the domain of ? The domain would be _________

52. What is the domain of ? The domain would be _________

53. Find the domain of each function:

54. Answers: