The document outlines the fundamentals of functions in calculus, including their definitions, representations, and properties. It discusses various ways to express functions, such as through formulas, numeric tables, graphs, and verbal descriptions, while also introducing the concepts of monotonicity and symmetry. The importance of functions in modeling real-world problems and their different types, such as piecewise-defined functions, is emphasized.

1. .
Sec on 1.1
Func ons and their Representa ons
V63.0121.001, Calculus I
Professor Ma hew Leingang
New York University
Announcements
First WebAssign-ments are due January 31
Do the Get-to-Know-You survey for extra credit!

2. Section 1.1
Functions and their
Representations
V63.0121.001, Calculus I
Professor Ma hew Leingang
New York University

3. Announcements
First WebAssign-ments
are due January 31
Do the Get-to-Know-You
survey for extra credit!

4. Objectives
Understand the defini on of
func on.
Work with func ons
represented in different ways
Work with func ons defined
piecewise over several intervals.
Understand and apply the
defini on of increasing and
decreasing func on.

5. What is a function?
Defini on
A func on f is a rela on which assigns to to every element x in a set
D a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { y | y = f(x) for some x } is called the range of f.

6. Outline
Modeling
Examples of func ons
Func ons expressed by formulas
Func ons described numerically
Func ons described graphically
Func ons described verbally
Proper es of func ons
Monotonicity
Symmetry

7. The Modeling Process
Real-world
.
. model Mathema cal
.
Problems Model
solve
test
Real-world interpret Mathema cal
. .
Predic ons Conclusions

8. Plato’s Cave
.

9. The Modeling Process
Real-world
.
. model Mathema cal
.
Problems Model
Shadows
Forms
solve
test
Real-world interpret Mathema cal
. .
Predic ons Conclusions

10. Outline
Modeling
Examples of func ons
Func ons expressed by formulas
Func ons described numerically
Func ons described graphically
Func ons described verbally
Proper es of func ons
Monotonicity
Symmetry

11. Functions expressed by formulas
Any expression in a single variable x defines a func on. In this case,
the domain is understood to be the largest set of x which a er
subs tu on, give a real number.

12. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2

13. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2
Solu on
The denominator is zero when x = 2, so the domain is all real numbers
except 2. We write:
domain(f) = { x | x ̸= 2 }

14. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2
Solu on
x+1 2y + 1
As for the range, we can solve y = =⇒ x = . So as
x−2 y−1
long as y ̸= 1, there is an x associated to y.
range(f) = { y | y ̸= 1 }

15. How did you get that?
x+1
start y=
x−2

16. How did you get that?
x+1
start y=
x−2
cross-mul ply y(x − 2) = x + 1

17. How did you get that?
x+1
start y=
x−2
cross-mul ply y(x − 2) = x + 1
distribute xy − 2y = x + 1

18. How did you get that?
x+1
start y=
x−2
cross-mul ply y(x − 2) = x + 1
distribute xy − 2y = x + 1
collect x terms xy − x = 2y + 1

19. How did you get that?
x+1
start y=
x−2
cross-mul ply y(x − 2) = x + 1
distribute xy − 2y = x + 1
collect x terms xy − x = 2y + 1
factor x(y − 1) = 2y + 1

20. How did you get that?
x+1
start y=
x−2
cross-mul ply y(x − 2) = x + 1
distribute xy − 2y = x + 1
collect x terms xy − x = 2y + 1
factor x(y − 1) = 2y + 1
2y + 1
divide x=
y−1

21. No-no’s for expressions
Cannot have zero in the
denominator of an
expression
Cannot have a nega ve
number under an even
root (e.g., square root)
Cannot have the
logarithm of a nega ve
number

22. Piecewise-defined functions
Example
Let
{
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f
and graph the func on.

23. Piecewise-defined functions
Example Solu on
Let The domain is [0, 2]. The graph
{ can be drawn piecewise.
x2 0 ≤ x ≤ 1;
f(x) = 2
3−x 1 < x ≤ 2.
1
Find the domain and range of f
and graph the func on. .
0 1 2

24. Piecewise-defined functions
Example Solu on
Let The domain is [0, 2]. The graph
{ can be drawn piecewise.
x2 0 ≤ x ≤ 1;
f(x) = 2
3−x 1 < x ≤ 2.
1
Find the domain and range of f
and graph the func on. .
0 1 2

25. Piecewise-defined functions
Example Solu on
Let The domain is [0, 2]. The graph
{ can be drawn piecewise.
x2 0 ≤ x ≤ 1;
f(x) = 2
3−x 1 < x ≤ 2.
1
Find the domain and range of f
and graph the func on. .
0 1 2

26. Piecewise-defined functions
Example Solu on
Let The domain is [0, 2]. The graph
{ can be drawn piecewise.
x2 0 ≤ x ≤ 1;
f(x) = 2
3−x 1 < x ≤ 2.
1
Find the domain and range of f
and graph the func on. .
0 1 2
The range is [0, 2).

27. Functions described numerically
We can just describe a func on by a table of values, or a diagram.

28. Functions defined by tables I
Example
Is this a func on? If so, what is
the range?
x f(x)
1 4
2 5
3 6

29. Functions defined by tables I
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 5 3 6
3 6

30. Functions defined by tables I
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 5 3 6
3 6

31. Functions defined by tables I
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 5 3 6
3 6

32. Functions defined by tables I
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 5 3 6
3 6

33. Functions defined by tables I
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 5 3 6
3 6
Yes, the range is {4, 5, 6}.

34. Functions defined by tables II
Example
Is this a func on? If so, what is
the range?
x f(x)
1 4
2 4
3 6

35. Functions defined by tables II
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 4 3 6
3 6

36. Functions defined by tables II
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 4 3 6
3 6

37. Functions defined by tables II
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 4 3 6
3 6

38. Functions defined by tables II
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 4 3 6
3 6

39. Functions defined by tables II
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
2 4 3 6
3 6
Yes, the range is {4, 6}.

40. Functions defined by tables III
Example
Is this a func on? If so, what is
the range?
x f(x)
1 4
1 5
3 6

41. Functions defined by tables III
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
1 5 3 6
3 6

42. Functions defined by tables III
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
1 5 3 6
3 6

43. Functions defined by tables III
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
1 5 3 6
3 6

44. Functions defined by tables III
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
1 5 3 6
3 6

45. Functions defined by tables III
Example Solu on
Is this a func on? If so, what is
the range? 1 . 4
x f(x)
2 5
1 4
1 5 3 6
3 6
This is not a func on.

46. An ideal function

47. An ideal function
Domain is the bu ons

48. An ideal function
Domain is the bu ons
Range is the kinds of soda
that come out

49. An ideal function
Domain is the bu ons
Range is the kinds of soda
that come out
You can press more than
one bu on to get some
brands

50. An ideal function
Domain is the bu ons
Range is the kinds of soda
that come out
You can press more than
one bu on to get some
brands
But each bu on will only
give one brand

51. Why numerical functions matter
Ques on
Why use numerical func ons at all? Formula func ons are so much
easier to work with.

52. Why numerical functions matter
Ques on
Why use numerical func ons at all? Formula func ons are so much
easier to work with.
Answer
In science, func ons are o en defined by data.
Or, we observe data and assume that it’s close to some nice
con nuous func on.

53. Numerical Function Example
Example
Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August 22–29, 2008.
100
90
80
70
60
50
40
30
20
10 .
8/22 8/23 8/24 8/25 8/26 8/27 8/28 8/29

54. Functions described graphically
Some mes all we have is the “picture” of a func on, by which we
mean, its graph.
The graph on the right represents a rela on but not a func on.

55. Functions described graphically
Some mes all we have is the “picture” of a func on, by which we
mean, its graph.
.
The graph on the right represents a rela on but not a func on.

56. Functions described graphically
Some mes all we have is the “picture” of a func on, by which we
mean, its graph.
. .
The graph on the right represents a rela on but not a func on.

57. Functions described graphically
Some mes all we have is the “picture” of a func on, by which we
mean, its graph.
. .
The graph on the right represents a rela on but not a func on.

58. Functions described graphically
Some mes all we have is the “picture” of a func on, by which we
mean, its graph.
. .
The graph on the right represents a rela on but not a func on.

59. Functions described verbally
O en mes our func ons come out of nature and have verbal
descrip ons:
The temperature T(t) in this room at me t.

60. Functions described verbally
O en mes our func ons come out of nature and have verbal
descrip ons:
The temperature T(t) in this room at me t.
The eleva on h(θ) of the point on the equator at longitude θ.

61. Functions described verbally
O en mes our func ons come out of nature and have verbal
descrip ons:
The temperature T(t) in this room at me t.
The eleva on h(θ) of the point on the equator at longitude θ.
The u lity u(x) I derive by consuming x burritos.

62. Outline
Modeling
Examples of func ons
Func ons expressed by formulas
Func ons described numerically
Func ons described graphically
Func ons described verbally
Proper es of func ons
Monotonicity
Symmetry

63. Monotonicity
Example
Let P(x) be the
probability that
my income was
at least $x last
year. What
might a graph of
P(x) look like?

64. Monotonicity
Example Solu on
Let P(x) be the
probability that 1
my income was
at least $x last
year. What 0.5
might a graph of
P(x) look like? .
$0 $52,115 $100K

65. Monotonicity
Defini on
A func on f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for
any two points x1 and x2 in the domain of f.
A func on f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2 for
any two points x1 and x2 in the domain of f.

66. Examples
Example
Going back to the burrito func on, would you call it increasing?

67. Examples
Example
Going back to the burrito func on, would you call it increasing?
Answer
Not if they are all consumed at once! Strictly speaking, the
insa ability principle in economics means that u li es are always
increasing func ons.

68. Examples
Example
Going back to the burrito func on, would you call it increasing?
Answer
Not if they are all consumed at once! Strictly speaking, the
insa ability principle in economics means that u li es are always
increasing func ons.
Example
Obviously, the temperature in Boise is neither increasing nor
decreasing.

69. Symmetry
Consider the following func ons described as words
Example
Let I(x) be the intensity of light x distance from a point.
Example
Let F(x) be the gravita onal force at a point x distance from a black
hole.
What might their graphs look like?

70. Possible Intensity Graph
Example Solu on
Let I(x) be the intensity
of light x distance from y = I(x)
a point. Sketch a
possible graph for I(x).
.
x

71. Possible Gravity Graph
Example Solu on
Let F(x) be the
gravita onal force at a y = F(x)
point x distance from a
black hole. Sketch a
possible graph for F(x). .
x

72. Definitions
Defini on
A func on f is called even if f(−x) = f(x) for all x in the domain
of f.
A func on f is called odd if f(−x) = −f(x) for all x in the
domain of f.

73. Examples
Example
Even: constants, even powers, cosine
Odd: odd powers, sine, tangent
Neither: exp, log

74. Summary
The fundamental unit of inves ga on in calculus is the func on.
Func ons can have many representa ons