No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
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Lesson 1: Functions
1. Section 1.1
Functions and their Representations
V63.0121.021/041, Calculus I
New York University
September 8, 2010
Announcements
First WebAssign-ments are due September 13
First written assignment is due September 15
Do the Get-to-Know-You survey for extra credit!
2. Announcements
First WebAssign-ments are
due September 13
First written assignment is
due September 15
Do the Get-to-Know-You
survey for extra credit!
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 2 / 33
4. Objectives: Functions and their Representations
Understand the definition of
function.
Work with functions
represented in different ways
Work with functions defined
piecewise over several
intervals.
Understand and apply the
definition of increasing and
decreasing function.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 3 / 33
5. What is a function?
Definition
A function f is a relation 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 .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 4 / 33
6. Outline
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 5 / 33
7. The Modeling Process
Real-world model Mathematical
Problems Model
solve
test
Real-world interpret Mathematical
Predictions Conclusions
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 6 / 33
9. The Modeling Process
Real-world model Mathematical
Problems Model
solve
test
Real-world interpret Mathematical
Predictions Conclusions
Shadows Forms
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 8 / 33
10. Outline
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 9 / 33
11. Functions expressed by formulas
Any expression in a single variable x defines a function. In this case, the
domain is understood to be the largest set of x which after substitution,
give a real number.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 10 / 33
12. Formula function example
Example
x +1
Let f (x) = . Find the domain and range of f .
x −2
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
13. Formula function example
Example
x +1
Let f (x) = . Find the domain and range of f .
x −2
Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
14. Formula function example
Example
x +1
Let f (x) = . Find the domain and range of f .
x −2
Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2. As for the range, we can solve
x +1 2y + 1
y= =⇒ x =
x −2 y −1
So as long as y = 1, there is an x associated to y .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
15. Formula function example
Example
x +1
Let f (x) = . Find the domain and range of f .
x −2
Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2. As for the range, we can solve
x +1 2y + 1
y= =⇒ x =
x −2 y −1
So as long as y = 1, there is an x associated to y . Therefore
domain(f ) = { x | x = 2 }
range(f ) = { y | y = 1 }
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
16. How did you get that?
x +1
start y=
x −2
cross-multiply 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
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 12 / 33
17. No-no’s for expressions
Cannot have zero in the
denominator of an
expression
Cannot have a negative
number under an even root
(e.g., square root)
Cannot have the logarithm
of a negative number
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 13 / 33
18. 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 function.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 14 / 33
19. 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 function.
Solution
The domain is [0, 2]. The range is [0, 2). The graph is piecewise.
2
1
0 1 2
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 14 / 33
20. Functions described numerically
We can just describe a function by a table of values, or a diagram.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 15 / 33
21. Example
Is this a function? If so, what is the range?
x f (x)
1 4
2 5
3 6
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
22. Example
Is this a function? If so, what is the range?
1 4
x f (x)
1 4 2 5
2 5
3 6
3 6
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
23. Example
Is this a function? If so, what is the range?
1 4
x f (x)
1 4 2 5
2 5
3 6
3 6
Yes, the range is {4, 5, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
24. Example
Is this a function? If so, what is the range?
x f (x)
1 4
2 4
3 6
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
25. Example
Is this a function? If so, what is the range?
1 4
x f (x)
1 4 2 5
2 4
3 6
3 6
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
26. Example
Is this a function? If so, what is the range?
1 4
x f (x)
1 4 2 5
2 4
3 6
3 6
Yes, the range is {4, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
27. Example
How about this one?
x f (x)
1 4
1 5
3 6
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
28. Example
How about this one?
1 4
x f (x)
1 4 2 5
1 5
3 6
3 6
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
29. Example
How about this one?
1 4
x f (x)
1 4 2 5
1 5
3 6
3 6
No, that one’s not “deterministic.”
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
31. An ideal function
Domain is the buttons
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
32. An ideal function
Domain is the buttons
Range is the kinds of soda
that come out
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
33. An ideal function
Domain is the buttons
Range is the kinds of soda
that come out
You can press more than one
button to get some brands
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
34. An ideal function
Domain is the buttons
Range is the kinds of soda
that come out
You can press more than one
button to get some brands
But each button will only
give one brand
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
35. Why numerical functions matter
In science, functions are often defined by data. Or, we observe data and
assume that it’s close to some nice continuous function.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 20 / 33
36. Numerical Function 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
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 21 / 33
37. Functions described graphically
Sometimes all we have is the “picture” of a function, by which we mean,
its graph.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 22 / 33
38. Functions described graphically
Sometimes all we have is the “picture” of a function, by which we mean,
its graph.
The one on the right is a relation but not a function.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 22 / 33
39. Functions described verbally
Oftentimes our functions come out of nature and have verbal descriptions:
The temperature T (t) in this room at time t.
The elevation h(θ) of the point on the equator at longitude θ.
The utility u(x) I derive by consuming x burritos.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 23 / 33
40. Outline
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 24 / 33
41. 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?
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 25 / 33
42. 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?
1
0.5
$0 $52,115 $100K
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 25 / 33
43. Monotonicity
Definition
A function f is decreasing if f (x1 ) > f (x2 ) whenever x1 < x2 for any
two points x1 and x2 in the domain of f .
A function f is increasing if f (x1 ) < f (x2 ) whenever x1 < x2 for any
two points x1 and x2 in the domain of f .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 26 / 33
44. Examples
Example
Going back to the burrito function, would you call it increasing?
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 27 / 33
45. Examples
Example
Going back to the burrito function, would you call it increasing?
Example
Obviously, the temperature in Boise is neither increasing nor decreasing.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 27 / 33
46. Symmetry
Example
Let I (x) be the intensity of light x distance from a point.
Example
Let F (x) be the gravitational force at a point x distance from a black hole.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 28 / 33
47. Possible Intensity Graph
y = I (x)
x
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 29 / 33
48. Possible Gravity Graph
y = F (x)
x
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 30 / 33
49. Definitions
Definition
A function f is called even if f (−x) = f (x) for all x in the domain of
f.
A function f is called odd if f (−x) = −f (x) for all x in the domain
of f .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 31 / 33
50. Examples
Even: constants, even powers, cosine
Odd: odd powers, sine, tangent
Neither: exp, log
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 32 / 33
51. Summary
The fundamental unit of investigation in calculus is the function.
Functions can have many representations
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 33 / 33