Lesson 1: Functions

Section 1.1
Functions and their Representations

V63.0121.021/041, Calculus I

New York University

September 7, 2010

Announcements

First WebAssign-ments are due September 14
Do the Get-to-Know-You survey for extra credit!

. . . . . .

Announcements

First WebAssign-ments are
due September 14
Do the Get-to-Know-You
survey for extra credit!

. . . . . .

V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 2 / 33

Function

. . . . . .

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
.
. m
. odel Mathematical
.
Problems Model

s
. olve
.est
t

. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions

. . . . . .


Plato's Cave

. . . . . .


The Modeling Process

. .
Real-world
.
. m
. odel Mathematical
.
Problems Model

s
. olve
.est
t

. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions

S
. hadows F
. orms
. . . . . .


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

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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

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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.”

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An ideal function

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An ideal function

Domain is the buttons

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An ideal function

Range is the kinds of soda
that come out

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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.
.
1
. 00 .
9
.0.
8
.0.
7
.0.
6
.0.
5
.0.
4
.0.
3
.0.
2
.0.
1
.0. . . . . . . .
8
. /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29
8 8 8 8 8 8 8

. . . . . .



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

. .5 .
0

. . .
$
.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?

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Examples

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

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

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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

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Summary

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

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