This document provides an overview of functions and continuity. It begins with essential questions about determining if functions are one-to-one and/or onto, and determining if functions are discrete or continuous. The document then defines key vocabulary terms related to functions, including one-to-one functions, onto functions, discrete relations, continuous relations, and more. It provides examples to demonstrate these concepts, such as evaluating functions, graphing equations, and determining if a relation represents a function.

1. Section 2-1
Functions and Continuity

2. Essential Questions
• How do you determine whether functions are one-to-
one and/or onto?
• How do you determine whether functions are discrete
or continuous?

3. Vocabulary
1. One-to-one Function:
2. Onto Function:
3. Discrete Relation:
4. Continuous Relation:

4. Vocabulary
1. One-to-one Function: When each element in the
domain pairs to exactly one unique element in the
range
2. Onto Function:
3. Discrete Relation:
4. Continuous Relation:

5. Vocabulary
1. One-to-one Function: When each element in the
domain pairs to exactly one unique element in the
range
2. Onto Function: When each element in the range
corresponds to an element in the domain
3. Discrete Relation:
4. Continuous Relation:

6. Vocabulary
1. One-to-one Function: When each element in the
domain pairs to exactly one unique element in the
range
2. Onto Function: When each element in the range
corresponds to an element in the domain
3. Discrete Relation: When the domain leads to a set of
unconnected points on a graph
4. Continuous Relation:

7. Vocabulary
1. One-to-one Function: When each element in the
domain pairs to exactly one unique element in the
range
2. Onto Function: When each element in the range
corresponds to an element in the domain
3. Discrete Relation: When the domain leads to a set of
unconnected points on a graph
4. Continuous Relation: When the domain leads to a
graph of connected points in a line or curve

8. Vocabulary
5. Vertical Line Test (VLT):
6. Independent Variable:
7. Dependent Variable:

9. Vocabulary
5. Vertical Line Test (VLT): If a vertical line is drawn on a
relation and can intersect the relation at more than
one spot for any x, then the relation is not a
function
6. Independent Variable:
7. Dependent Variable:

10. Vocabulary
5. Vertical Line Test (VLT): If a vertical line is drawn on a
relation and can intersect the relation at more than
one spot for any x, then the relation is not a
function
6. Independent Variable: The input value of a relation;
usually x
7. Dependent Variable:

11. Vocabulary
5. Vertical Line Test (VLT): If a vertical line is drawn on a
relation and can intersect the relation at more than
one spot for any x, then the relation is not a
function
6. Independent Variable: The input value of a relation;
usually x
7. Dependent Variable: The output value of a relation;
usually y

12. Vocabulary
8. Function Notation:

13. Vocabulary
8. Function Notation: Rewriting an equation by
replacing the dependent variable with f(x); This
identifies what is getting plugged in for the
independent variable

14. Vocabulary
8. Function Notation: Rewriting an equation by
replacing the dependent variable with f(x); This
identifies what is getting plugged in for the
independent variable
y = 3x − 7

15. Vocabulary
8. Function Notation: Rewriting an equation by
replacing the dependent variable with f(x); This
identifies what is getting plugged in for the
independent variable
y = 3x − 7

16. Vocabulary
8. Function Notation: Rewriting an equation by
replacing the dependent variable with f(x); This
identifies what is getting plugged in for the
independent variable
y = 3x − 7
f (x ) = 3x − 7

17. Example 1
State the domain and range of the relation. Then
determine whether the relation is a function. If it is a
function, determine if it is one-to-one, onto, both, or
neither.
x
y

18. Example 1
State the domain and range of the relation. Then
determine whether the relation is a function. If it is a
function, determine if it is one-to-one, onto, both, or
neither.
x
y
D = {-4, -3, 0, 1, 3}

19. Example 1
State the domain and range of the relation. Then
determine whether the relation is a function. If it is a
function, determine if it is one-to-one, onto, both, or
neither.
x
y
D = {-4, -3, 0, 1, 3} R = {-2, 1, 2, 3}

20. Example 1
State the domain and range of the relation. Then
determine whether the relation is a function. If it is a
function, determine if it is one-to-one, onto, both, or
neither.
x
y
D = {-4, -3, 0, 1, 3} R = {-2, 1, 2, 3}
This is a function (passes VLT).

21. Example 1
State the domain and range of the relation. Then
determine whether the relation is a function. If it is a
function, determine if it is one-to-one, onto, both, or
neither.
x
y
D = {-4, -3, 0, 1, 3} R = {-2, 1, 2, 3}
This is onto, but not one-to-one. Each
range value matches to a domain
value, but each domain value does not
match to a unique range value.
This is a function (passes VLT).

22. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y

23. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

24. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

25. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

26. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

27. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

28. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

29. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

30. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4

31. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4
D = !

32. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4
D = !
R = !

33. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4
D = !
R = !
Function

34. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4
D = !
R = !
Function
Both one-to-one
and onto

35. Example 2
Graph the equation and determine the domain and
range. Then determine whether the equation is a
function, is one-to-one, onto, both, or neither. State
whether it is discrete or continuous.
x
y
y =
2
3
x + 4
D = !
R = !
Function
Both one-to-one
and onto
Continuous

36. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
a. f (−2)

37. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
a. f (−2)
f (−2) = (−2)3
− 3

38. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
a. f (−2)
f (−2) = (−2)3
− 3
f (−2) = −8 − 3

39. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
a. f (−2)
f (−2) = (−2)3
− 3
f (−2) = −8 − 3
f (−2) = −11

40. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
b. f (2t )

41. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
b. f (2t )
f (2t ) = (2t )3
− 3

42. Example 3
Given the function, find each value.
f (x ) = x 3
− 3
b. f (2t )
f (2t ) = (2t )3
− 3
f (2t ) = 8t 3
− 3

43. Example 4
The table shows the average fuel efficiency in miles per
gallon for SUVs for several years. Graph this information
and determine whether it represents a function. Is the
relation discrete or continuous?

44. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6

45. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6

46. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year

47. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
2001
2002
2003
2004
2005
2006
2007

48. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg
2001
2002
2003
2004
2005
2006
2007

49. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

50. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

51. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

52. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

53. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

54. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

55. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

56. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007

57. Example 4
Year Fuel Efficiency (mpg)
2001 20.8
2002 20.6
2003 20.8
2004 20.9
2005 21.6
2006 22.3
2007 22.6
Year
mpg20.6
21.0
21.4
21.8
22.2
22.6
20.2
2001
2002
2003
2004
2005
2006
2007
This is a discrete function

58. Example 5
A commuter train ticket costs $7.25. The cost of taking
the train x times can be described by the function
y = 7.25x, where y is the total cost in dollars. Determine
whether the function is correctly modeled by a discrete
or continuous function. Explain your reasoning.

59. Example 5
A commuter train ticket costs $7.25. The cost of taking
the train x times can be described by the function
y = 7.25x, where y is the total cost in dollars. Determine
whether the function is correctly modeled by a discrete
or continuous function. Explain your reasoning.
This is a discrete function as each trip is one ticket. You
cannot purchase a partial ticket, so the domain is the
set of whole numbers