Lesson 1: Functions

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!

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

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.


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 .


Outline

Modeling

Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally

Properties of functions
Monotonicity
Symmetry


The Modeling Process

Real-world model Mathematical
Problems Model

solve
test

Real-world interpret Mathematical
Predictions Conclusions


Plato’s Cave


The Modeling Process

Real-world model Mathematical
Problems Model

solve
test

Real-world interpret Mathematical
Predictions Conclusions

Shadows Forms


Outline

Modeling


Monotonicity
Symmetry



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.


Formula function example

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



Example
x +1
x −2

Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2.



Example
x +1
x −2

Solution
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 .



Example
x +1
x −2

Solution
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 }


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


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


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.


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



We can just describe a function by a table of values, or a diagram.


Example

Is this a function? If so, what is the range?

x f (x)
1 4
2 5
3 6


Example


1 4
x f (x)
1 4 2 5
2 5
3 6
3 6


Example


1 4
x f (x)
1 4 2 5
2 5
3 6
3 6

Yes, the range is {4, 5, 6}.


Example


x f (x)
1 4
2 4
3 6


Example


1 4
x f (x)
1 4 2 5
2 4
3 6
3 6


Example


1 4
x f (x)
1 4 2 5
2 4
3 6
3 6

Yes, the range is {4, 6}.


Example

How about this one?

x f (x)
1 4
1 5
3 6


Example

How about this one?

1 4
x f (x)
1 4 2 5
1 5
3 6
3 6


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.”


An ideal function


An ideal function

Domain is the buttons


An ideal function

Range is the kinds of soda
that come out


An ideal function

that come out
You can press more than one
button to get some brands


An ideal function

that come out
You can press more than one
button to get some brands
But each button will only
give one brand


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.


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



Sometimes all we have is the “picture” of a function, by which we mean,
its graph.



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.



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.


Outline

Modeling


Monotonicity
Symmetry


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?


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


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 .


Examples

Example
Going back to the burrito function, would you call it increasing?


Examples

Example
Going back to the burrito function, would you call it increasing?

Example
Obviously, the temperature in Boise is neither increasing nor decreasing.


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.


Possible Intensity Graph

y = I (x)

x


Possible Gravity Graph

y = F (x)

x


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 .


Examples

Even: constants, even powers, cosine
Odd: odd powers, sine, tangent
Neither: exp, log


Summary

The fundamental unit of investigation in calculus is the function.
Functions can have many representations


Lesson 1: Functions

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Lesson 1: Functions