1. Answer:
-3 if x ≥ -4
f(x) =
-x if x ≤ -1

2. Answer:
-1 if x ≥ 2
f(x) =
x2 if x < 2

3. Set up your notes – Topic is “Functions”
Wed, 10/31
Happy Halloween!
SWBAT… define a function, learn function notation, and a evaluate function
Agenda
Lots of notes on functions with many practice problems (40 min)
OYO Problems (10 min)
Warm-Up:
Set up your notes – Topic is “Functions”
HW#6: Page 1

4. Ms. Sophia Papaefthimiou
Infinity HS

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

6. 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.

7. Function Notation
The most common name is "f", but you can have other names like "g"
What goes into the function (the input) 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?
Answer: g(a) 7

8. Function Notation: The Symbolic Form
Name of the function
• Input Value • Domain • Independent Variable
• Output Value • Range • Dependent Variable
One of the really big deals is to remember that y is f(x). That means that f(x) and y are interchangeable.
The output is y = f(x)

9. Function Notation y f(x) f(x) = 3x + 2
Function notation replaces the ___ in an equation with ___
Example: Given y = 3x + 2, write the equation in function notation
f(x) = 3x + 2
Question: Write y = x2 in function notation.
f(x) = x2
Answer: f(x) = x2 9

10. Function Notation y = 2x + 3 Now you say “f(x) = 2x + 3; find f(-1)”
You used to say “y = 2x + 3; find the value of y when x = -1”
y = 2x + 3
y = 2(-1) + 3
y =
y = 1
Now you say “f(x) = 2x + 3; find f(-1)”
f(x) = 2x + 3 find f(-1)
f(-1) = 2(-1) + 3
f(-1) =
f(-1) = 1 -1 1
f(-1) = 2x + 3 10

11. A function P is defined as follows:
For x > 0, P(x) = x5 + x4 – 36x – 36
For x < 0, P(x) = -x5 + x4 + 36x – 36
What is the value of P(-1)?
A. -70
B. -36
C. 0
D. 36

12. Evaluating functions:
Directions:
If f(x) = 2x – 4 and g(x) = x² – 4x, find each value:
f(-3)
f(3x)
g(t)
f(q + 1)
f(2) + g(-2)
f(g(-2)) (Hint: Start from the inside out. Find g(-2) first)
f(x) = 2x – 4
f(-3) = 2(-3) – 4
f(-3) = -6 – 4
f(-3) = -10
f(3x) = 2(3x) – 4
f(3x) = 6x – 4
3. g(x) = x2 – 4x
g(t) = t2 – 4t
f(q + 1) = 2(q + 1) – 4
f(q + 1) = 2q + 2 – 4
f(q + 1) = 2q – 2
5. f(x) = 2x – 4 g(x) = x2 – 4x
f(2) = 2(2) – 4 g(-2) = (-2)2 – 4(-2)
f(2) = 4 – 4 g(-2) = 4 + 8
f(2) = 0 g(-2) = 12
f(2) + g(-2) = 12
6. f(g(-2)) (from #5 g(-2) = 12)
f(12) = 2(12) – 4
f(12) = 24 – 4
f(12) = 20

13. Revisit our objectives
Today:
To define a function
To learn function notation
To evaluate functions

14. SWBAT… use the vertical line test
Fri, 11/2
SWBAT… use the vertical line test
Agenda
WU (20 min)
Review HW#6 – page 1 (25 min)
Warm-Up:
Let f(t) be the number of people, in millions, who own cell phones t years after Explain the meaning of the following statements.
1. f(10) = 100.3
2. f(a) = 20
3. f(20) = b
4. n = f(t)
HW#6: Functions (page 1 – 4)

15. Solution: Cell phones
1. f(10) = 100.3: The number of people who own cell phones in the year 2000 is 100,300, f(a) = 20: There are 20,000,000 people who own cell phones a years after f(20) = b: There will be b million people who own cell phones in the year n = f(t): t years after 1990, there will be n million people who own cell phones.

17. Objectives Today To learn function mapping
To conduct the vertical line test
To find the domain and range of a function
To write a function as an ordered pair

18. Function Mapping
A set of points or equation where every input has exactly one output.
In other words, the domain or x value can not be repeated
“Exactly one" means that a function is single valued. It will not give back 2 or more results for the same input.
So, "f(2) = 7 or f(2) = 9" is not a function! 18

19. Function Mapping (cont’d)
This is a function!
There is only one arrow coming from 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.

20. Function Mapping (cont’d)
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.

21. 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.

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

23. Domain and Range
• Domain: What can go into a function. The set of all x values in a function. How “wide” the function is.
• Range: What comes out of a function. The set of all y values in a function. How “tall” the function is.

24. The domain is the set of all real numbers.
The range is y ≥ 0.

25. SWBAT… list the domain and range of functions
Mon, 11/5
SWBAT… list the domain and range of functions
Agenda
WU (10 min)
Practice: Evaluating functions (15 min)
Review HW#6 – Page 1 – 3 (25 min)
Warm-Up:
Write your HW in your planner.
Take out your HW#6-Functions packet
What is the domain and range of y = x + 1?
HW#6: Functions (see agenda problems)

26. What is the domain and range of y = x + 1?
Domain: All real numbers
Range: All real numbers
All real numbers.

27. Domain and Range (from Friday)
f(x) = x2 – 2
The domain is the set of all real numbers.
The range is y ≥ -2.

28. Find the domain and range of f(x) = ½ x2-4
f(-4) = 8 -2
f(-2) = 2 -1
f(-1) = 0.5
f(0) = 0 1
f(1) = 0.5 2
f(2) = 2 4
f(4) = 8
The domain is all real numbers.
Explanation: There are no restrictions on the domain, the x value.
The range is y ≥ 0.
Explanation: The graph will never be below 0.

29. Review HW#6

30. f(x) = -3x and g(x) = 2x
Solve f(x) = 0
Solve f(x) > 0
Solve f(x) = g(x)
Solve f(x) < g(x)

31. f(x) = -3x + 10 and g(x) = 2x Solve f(x) = 0 Solve f(x) > 0
Solve f(x) = g(x)
x = 2
Solve f(x) < g(x)
x > 2

32. HW#6 – Page 1 f(4) = 4 g(2) = -4 g(-3) = 21 f(-5) = -14 f(3x) =6x – 4
g(t) = t2 – 4t
f(h) = 2h – 4
f(q + 1) = 2q – 2
f(2) + g(-2) = 12
g(-b) = b2 + 4b
f(r – 1) = 2r – 6

33. HW – Due Tomorrow 1.) Given f(x) = x2 – 1: Graph the function
Find the domain
Find the range
2.)Given f(x) = x – 1:

34. Question 1/10
For the function f(x) = x2, if the domain is {1, 2, 3}, what is the range?
Answer: Range = {1, 4, 9} 34

35. Question 2/10
Given f(x) = 3x – 5 and the domain is {0, 2, -1} find the range
f(x) = 3x – 5
f(0) = 3(0) – 5 = -5
f(2) = 3(2) – 5 = 1
f(1) = 3(-1) – 5 = -8
Range = {-5, 1, -8} 35

36. Question 3/10 If f(x) = -x2 find a. f(3) b. f(-3)
a.) f(x) = -x2 b.) f(x) = -x2
f(3) = -(3) f(-3) = -(-3)2
f(3) = -9 f(-3) = -9
For both of these problems, you first have to calculate the exponents, then find the negative. 36

37. Question 4/10 Function f is defined by f(x) = -2x2 + 6x – 3 Find f(-2)
Write as an ordered pair
Substitute x with -2 in the formula of the function and calculate f(-2) as follows f(-2) = -2(-2)2 + 6(-2) – 3 f(-2) = (-2)(4) – 3
– 3 = -23
This means (-2, -23) is a point in the function 37

38. Question 5/10 Does the diagram represent a function? Why? 1 7 24 3 151 7 24 3 15
Yes, because no x-value is repeated. 38

39. Question 6/10 Suppose g(x) = 2x and f(x) = 4x. What is g(5) + f(-9)?
Answer: g(5) + f(-9) = -26 39

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

41. 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

42. 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.

43. 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)

44. Review HW#6 – Page 1

45. HW#6 – Page 1 f(4) = 4 g(2) = -4 g(-3) = 21 f(-5) = -14 f(3x) =6x – 4
g(t) = t2 – 4t
f(h) = 2h – 4
f(q + 1) = 2q – 2
f(2) + g(-2) = 12
g(-b) = b2 + 4b
f(r – 1) = 2r – 6

46. HW#6 – Page 1
13.

47. SWBAT… find the domain and range of functions
Mon, 11/5
SWBAT… find the domain and range of functions
Agenda
WU (15 min)
Domain & range practice problems (20 min)
Warm-Up:
HW#6: Functions – Page 5 – 6

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

49. When finding the domain, remember:
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:
The denominator of a fraction cannot be zero
The values under a square root sign must be positive
All real numbers.

50. Find the domain for the function f(x) = x2 + 2. Explain.
The domain is “all real numbers of x” because there are no restrictions on the value of x.

51. When finding the domain, remember:
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:
Substitute different x-values into the expression to see what is happening for y.
Draw a sketch! In math, it's very true that a picture is worth a thousand words
All real numbers.

52. Find the range for the function f(x) = x2 – 5. Explain.
The range is “y ≥ -5” since x2 – 5 is never less than -5.

53. What is the domain and range of f(x) = x2 + 3? Explain.
Domain: All real numbers because there are no restrictions on x.
Range: y ≥ 3 because x2 + 3 is less than 3.
All real numbers.

54. What is the domain of ?

55. What is the range of ?

56. What is the domain and range of ?
Domain: x ≥ 1
Range: y ≥ 0
All real numbers.

57. What is the domain and range of y = x + 1?
Domain: All real numbers
Range: All real numbers
All real numbers.

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

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

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

61. The colon says, such that

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

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

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

65. What is the domain of ? The domain would be _________
Answer: All numbers except 0.
“The set of x such that x does not equal 0.” 65

66. What is the domain of ?
The domain would be ___________

67. What is the domain of ?
The domain would be __________

68. What is the domain of y = -(x – 4)?
The domain would be __________

69. Find the domain of each function:

70. Answers: