This document provides an introduction to functions and key function concepts like domain, range, independent and dependent variables. It begins with examples of identifying independent and dependent variables in equations and tables. It then defines a function and discusses how to determine if a set of ordered pairs forms a function. The document concludes by discussing the domain and range of functions.

1. WARM-UP:
1. [18 ÷ (2 + 1)] − 4
2. What is the difference of 10 and 1?
3. What is the quotient of 35 and 5?

2. WARM-UP:
1. [18 ÷ (2 + 1)] − 4
[18 ÷ 3] − 4
€
2. What is the difference of 10 and 1?
3. What is the quotient of 35 and 5?

3. WARM-UP:
1. [18 ÷ (2 + 1)] − 4
[18 ÷ 3] − 4
[6] − 4
€
€2. What is the difference of 10 and 1?
3. What is the quotient of 35 and 5?

4. WARM-UP:
1. [18 ÷ (2 + 1)] − 4
[18 ÷ 3] − 4
[6] − 4
2
€
€2. What is the difference of 10 and 1?
€
3. What is the quotient of 35 and 5?

5. WARM-UP:
1. [18 ÷ (2 + 1)] − 4
[18 ÷ 3] − 4
[6] − 4
2
€
€2. What is the difference of 10 and 1?
10 −1 = 9
€
3. What is the quotient of 35 and 5?
€

6. WARM-UP:
1. [18 ÷ (2 + 1)] − 4
[18 ÷ 3] − 4
[6] − 4
2
€
€2. What is the difference of 10 and 1?
10 −1 = 9
€
3. What is the quotient of 35 and 5?
€ 35 ÷ 5 = 7

7. 1.2 WHAT IS A
FUNCTION?

8. ESSENTIAL QUESTION:
How do we determine if a set of ordered pairs or
table is a function?

9. VOCABULARY:
Dependent Variable:
Independent Variable:
Function:

10. VOCABULARY:
Dependent Variable: relies on another variable
Independent Variable:
Function:

11. VOCABULARY:
Dependent Variable: relies on another variable
usually “y” variable
Independent Variable:
Function:

12. VOCABULARY:
Dependent Variable: relies on another variable
usually “y” variable
Independent Variable: does not rely on another variable
Function:

13. VOCABULARY:
Dependent Variable: relies on another variable
usually “y” variable
Independent Variable: does not rely on another variable
usually the “x” variable
Function:

14. VOCABULARY:
Dependent Variable: relies on another variable
usually “y” variable
Independent Variable: does not rely on another variable
usually the “x” variable
Function:
correspondence or pairing between two variables such
that each value of the 1st (independent) variable
corresponds to exactly one value of the 2nd
(dependent) variable

15. EXAMPLE
1. The equation h = 2t gives the number of inches h of new
snow after t hours if snow falls during a storm at the rate of 2
inches per hour.
Identify the independent and dependent variables.
Independent Variable:
Dependent Variable:

16. EXAMPLE
1. The equation h = 2t gives the number of inches h of new
snow after t hours if snow falls during a storm at the rate of 2
inches per hour.
Identify the independent and dependent variables.
Independent Variable: t - time in hours
Dependent Variable:

17. EXAMPLE
1. The equation h = 2t gives the number of inches h of new
snow after t hours if snow falls during a storm at the rate of 2
inches per hour.
Identify the independent and dependent variables.
Independent Variable: t - time in hours
Dependent Variable: h - inches of snow

18. The dependent variable ______________ the independent variable.

19. The dependent variable ______________ the independent variable.
is a function of

20. The dependent variable ______________ the independent variable.
is a function of
y

21. The dependent variable ______________ the independent variable.
is a function of
y x

22. The dependent variable ______________ the independent variable.
is a function of
y x
In your graphing calculator type y = 3x + 7

23. The dependent variable ______________ the independent variable.
is a function of
y x
In your graphing calculator type y = 3x + 7
Look at the table of values.
What is the input and what is the output?

24. The dependent variable ______________ the independent variable.
is a function of
y x
In your graphing calculator type y = 3x + 7
Look at the table of values.
What is the input and what is the output?
Input - values for x

25. The dependent variable ______________ the independent variable.
is a function of
y x
In your graphing calculator type y = 3x + 7
Look at the table of values.
What is the input and what is the output?
Input - values for x
Output - values for y

26. The table shows the average temperature T in degrees
Fahrenheit for each month M in Honolulu, Hawaii.
M T
Jan. 73
2. Is T a function of M?
Feb. 73
March 74
April 76 3. Is M a function of T?
May 78
June 79
July 80
August 81
Sept. 81
Oct. 80
Nov. 77
Dec. 74

27. The table shows the average temperature T in degrees
Fahrenheit for each month M in Honolulu, Hawaii.
M T
Jan. 73
2. Is T a function of M? YES
Feb. 73
March 74
April 76 3. Is M a function of T?
May 78
June 79
July 80
August 81
Sept. 81
Oct. 80
Nov. 77
Dec. 74

28. The table shows the average temperature T in degrees
Fahrenheit for each month M in Honolulu, Hawaii.
M T
Jan. 73
2. Is T a function of M? YES
Feb. 73
March 74
April 76 3. Is M a function of T? NO
May 78
June 79
July 80
August 81
Sept. 81
Oct. 80
Nov. 77
Dec. 74

29. The table shows the average temperature T in degrees
Fahrenheit for each month M in Honolulu, Hawaii.
M T
Jan. 73
2. Is T a function of M? YES
Feb. 73
March 74
April 76 3. Is M a function of T? NO
May 78
June 79
July 80
August 81 What is the difference between
Sept. 81
Oct. 80 the 2 questions?
Nov. 77
Dec. 74

30. The table shows the average temperature T in degrees
Fahrenheit for each month M in Honolulu, Hawaii.
M T
Jan. 73
2. Is T a function of M? YES
Feb. 73
March 74
April 76 3. Is M a function of T? NO
May 78
June 79
July 80
August 81 What is the difference between
Sept. 81
Oct. 80 the 2 questions?
Nov. 77
Dec. 74
According to the definition; the 1st variable can only
correspond to 1 value of the 2nd variable. ie the second
variable can not be listed twice.

31. EXAMPLE:
s 1 1 2 2 3 3
r 3 -3 6 -6 9 -9
4. Is r a function of s?

32. EXAMPLE:
s 1 1 2 2 3 3
r 3 -3 6 -6 9 -9
4. Is r a function of s?
No; each s-value is not paired with exactly 1 r-value
OR
s has repeated values

33. EXAMPLE:
s 1 1 2 2 3 3
r 3 -3 6 -6 9 -9
4. Is r a function of s?
No; each s-value is not paired with exactly 1 r-value
OR
s has repeated values
5. Is s a function of r?

34. EXAMPLE:
s 1 1 2 2 3 3
r 3 -3 6 -6 9 -9
4. Is r a function of s?
No; each s-value is not paired with exactly 1 r-value
OR
s has repeated values
5. Is s a function of r?
Yes; every r-value is paired with exactly 1 s-value
OR
r does not have repeated values

35. EXAMPLE:
6. The table gives the high school enrollment, in millions, in
the United States from 1985 to 1991.
Is the female enrollment a function of the year?
Year Male Female
1985 7.2 6.9
1986 7.2 7.0
1987 7.0 6.8
1988 6.7 6.4
1989 6.6 6.3
1990 6.5 6.4
1991 6.8 6.4

36. EXAMPLE:
6. The table gives the high school enrollment, in millions, in
the United States from 1985 to 1991.
Is the female enrollment a function of the year?
Year Male Female
1985 7.2 6.9 Yes;
1986 7.2 7.0
each year is paired with exactly 1
female enrollment figure
1987 7.0 6.8
OR
1988 6.7 6.4
the year does not repeat itself
1989 6.6 6.3
1990 6.5 6.4
1991 6.8 6.4

37. VOCABULARY
CONTINUED...
Domain of a Function:
Range of a Function:

38. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:

39. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:
set of values of the dependent variable that can result from
the substitution for the independent variable (y-values)
(OUTPUT)

40. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:
set of values of the dependent variable that can result from
the substitution for the independent variable (y-values)
(OUTPUT)
Refer to the temperature example:

41. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:
set of values of the dependent variable that can result from
the substitution for the independent variable (y-values)
(OUTPUT)
Refer to the temperature example:
Domain:

42. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:
set of values of the dependent variable that can result from
the substitution for the independent variable (y-values)
(OUTPUT)
Refer to the temperature example:
Domain:
Range:

43. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:
set of values of the dependent variable that can result from
the substitution for the independent variable (y-values)
(OUTPUT)
Refer to the temperature example:
Domain: set of months in a year
Range:

44. VOCABULARY
CONTINUED...
Domain of a Function:
set of values, which are allowable substitutions for the
independent variable (x-values) (INPUT)
Range of a Function:
set of values of the dependent variable that can result from
the substitution for the independent variable (y-values)
(OUTPUT)
Refer to the temperature example:
Domain: set of months in a year
Range: {73, 74, 76, 77, 78, 79, 80, 81}

45. EXAMPLE:
7. If y is a function of x, what real numbers are not in the
1
domain of y = 2 ?
x − 64
€

46. EXAMPLE:
7. If y is a function of x, what real numbers are not in the
1
domain of y = 2 ?
x − 64
Clue: Is there a value for x that I can not have?
€

47. EXAMPLE:
7. If y is a function of x, what real numbers are not in the
1
domain of y = 2 ?
x − 64
Clue: Is there a value for x that I can not have?
8 €

48. EXAMPLE:
7. If y is a function of x, what real numbers are not in the
1
domain of y = 2 ?
x − 64
Clue: Is there a value for x that I can not have?
8 € and

49. EXAMPLE:
7. If y is a function of x, what real numbers are not in the
1
domain of y = 2 ?
x − 64
Clue: Is there a value for x that I can not have?
8 € and -8

50. EXAMPLE:
7. If y is a function of x, what real numbers are not in the
1
domain of y = 2 ?
x − 64
Clue: Is there a value for x that I can not have?
8 € and -8
If x is 8 or -8 then the denominator is 0. Everyone knows we
can’t divide by 0. Therefore we can not have 8 and -8 as a
value for x.

51. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: Range:

52. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range:

53. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}

54. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order

55. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3

56. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3
For this let’s examine the graph

57. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3
For this let’s examine the graph

58. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3
For this let’s examine the graph
Domain:

59. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3
For this let’s examine the graph
Domain: all real numbers

60. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3
For this let’s examine the graph
Domain: all real numbers Range:

61. 8. What is the domain and range of
{(2, 4), (7, 11), (9, 13), (8, -4)}
Domain: {2, 7, 8, 9} Range: {-4, 4, 11, 13}
Notice: the numbers are listed in ascending order
9. What is the domain and range of y = x4 - 3
For this let’s examine the graph
Domain: all real numbers Range: {y : y ≥ −3}

62. SETS OF NUMBERS
* often used for the domains
Natural Numbers:
Whole Numbers:
Integers:
Rational Numbers:
Real Numbers:

63. SETS OF NUMBERS
* often used for the domains
Natural Numbers: a.k.a. ~ counting numbers
Whole Numbers:
Integers:
Rational Numbers:
Real Numbers:

64. SETS OF NUMBERS
* often used for the domains
Natural Numbers: a.k.a. ~ counting numbers
{1, 2, 3, 4, 5, 6, ...}
Whole Numbers:
Integers:
Rational Numbers:
Real Numbers:

65. SETS OF NUMBERS
* often used for the domains
Natural Numbers: a.k.a. ~ counting numbers
{1, 2, 3, 4, 5, 6, ...}
Whole Numbers: {0, 1, 2, 3, 4, 5,...}
Integers:
Rational Numbers:
Real Numbers:

66. SETS OF NUMBERS
* often used for the domains
Natural Numbers: a.k.a. ~ counting numbers
{1, 2, 3, 4, 5, 6, ...}
Whole Numbers: {0, 1, 2, 3, 4, 5,...}
Integers: {...-3, -2, -1, 0, 1, 2, 3,...}
Rational Numbers:
Real Numbers:

67. SETS OF NUMBERS
* often used for the domains
Natural Numbers: a.k.a. ~ counting numbers
{1, 2, 3, 4, 5, 6, ...}
Whole Numbers: {0, 1, 2, 3, 4, 5,...}
Integers: {...-3, -2, -1, 0, 1, 2, 3,...}
Rational Numbers:
numbers that can be represented as a ratio;
a/b where b can’t be 0.
Real Numbers:

68. SETS OF NUMBERS
* often used for the domains
Natural Numbers: a.k.a. ~ counting numbers
{1, 2, 3, 4, 5, 6, ...}
Whole Numbers: {0, 1, 2, 3, 4, 5,...}
Integers: {...-3, -2, -1, 0, 1, 2, 3,...}
Rational Numbers:
numbers that can be represented as a ratio;
a/b where b can’t be 0.
Real Numbers:
set of numbers represented by decimals (all numbers known
to YOU currently 0, -7.2, pi

69. EXAMPLES:
Give an example that will satisfy each of the conditions.
10. an integer that is not a natural number

70. EXAMPLES:
Give an example that will satisfy each of the conditions.
10. an integer that is not a natural number
-5 - any negative number

71. EXAMPLES:
Give an example that will satisfy each of the conditions.
10. an integer that is not a natural number
-5 - any negative number
11. a real number that is not an integer

72. EXAMPLES:
Give an example that will satisfy each of the conditions.
10. an integer that is not a natural number
-5 - any negative number
11. a real number that is not an integer
pi, .9 - any fraction

73. EXAMPLES:
Give an example that will satisfy each of the conditions.
10. an integer that is not a natural number
-5 - any negative number
11. a real number that is not an integer
pi, .9 - any fraction
12. an integer that is not a real number

74. EXAMPLES:
Give an example that will satisfy each of the conditions.
10. an integer that is not a natural number
-5 - any negative number
11. a real number that is not an integer
pi, .9 - any fraction
12. an integer that is not a real number
not possible

75. HOMEWORK:
page 15 #1-14 and 20-31