lesson 1: functions and their representations (slides)
DESCRIPTION
The function is the fundamental unit in calculus. There are many ways to describe functions: with words, pictures, symbols, or numbers.TRANSCRIPT
..
Sec on 1.1Func ons and their Representa ons
V63.0121.001, Calculus IProfessor Ma hew Leingang
New York University
Announcements
I First WebAssign-ments are due January 31I Do the Get-to-Know-You survey for extra credit!
Section 1.1Functions and theirRepresentations
V63.0121.001, Calculus IProfessor Ma hew Leingang
New York University
Announcements
I First WebAssign-mentsare due January 31
I Do the Get-to-Know-Yousurvey for extra credit!
ObjectivesI Understand the defini on of
func on.I Work with func onsrepresented in different ways
I Work with func ons definedpiecewise over several intervals.
I Understand and apply thedefini on of increasing anddecreasing func on.
What is a function?
Defini onA func on f is a rela on which assigns to to every element x in a setD a single element f(x) in a set E.
I The set D is called the domain of f.I The set E is called the target of f.I The set { y | y = f(x) for some x } is called the range of f.
OutlineModeling
Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally
Proper es of func onsMonotonicitySymmetry
The Modeling Process
...Real-worldProblems
.. Mathema calModel
.
.
Mathema calConclusions
.
.
Real-worldPredic ons
. model.
solve
.
interpret
.
test
Plato’s Cave
..
The Modeling Process
...Real-worldProblems
.. Mathema calModel
.
.
Mathema calConclusions
.
.
Real-worldPredic ons
. model.
solve
.
interpret
.
test
.Sh
adow
s.Form
s
OutlineModeling
Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally
Proper es of func onsMonotonicitySymmetry
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 ersubs tu on, give a real number.
Formula function exampleExample
Let f(x) =x+ 1x− 2
. Find the domain and range of f.
Solu on
Formula function exampleExample
Let f(x) =x+ 1x− 2
. Find the domain and range of f.
Solu onThe denominator is zerowhen x = 2, so the domain is all real numbersexcept 2. We write:
domain(f) = { x | x ̸= 2 }
Formula function exampleExample
Let f(x) =x+ 1x− 2
. Find the domain and range of f.
Solu on
As for the range, we can solve y =x+ 1x− 2
=⇒ x =2y+ 1y− 1
. So as
long as y ̸= 1, there is an x associated to y.
range(f) = { y | y ̸= 1 }
How did you get that?
start y =x+ 1x− 2
cross-mul ply y(x− 2) = x+ 1distribute xy− 2y = x+ 1
collect x terms xy− x = 2y+ 1factor x(y− 1) = 2y+ 1
divide x =2y+ 1y− 1
How did you get that?
start y =x+ 1x− 2
cross-mul ply y(x− 2) = x+ 1
distribute xy− 2y = x+ 1collect x terms xy− x = 2y+ 1
factor x(y− 1) = 2y+ 1
divide x =2y+ 1y− 1
How did you get that?
start y =x+ 1x− 2
cross-mul ply y(x− 2) = x+ 1distribute xy− 2y = x+ 1
collect x terms xy− x = 2y+ 1factor x(y− 1) = 2y+ 1
divide x =2y+ 1y− 1
How did you get that?
start y =x+ 1x− 2
cross-mul ply y(x− 2) = x+ 1distribute xy− 2y = x+ 1
collect x terms xy− x = 2y+ 1
factor x(y− 1) = 2y+ 1
divide x =2y+ 1y− 1
How did you get that?
start y =x+ 1x− 2
cross-mul ply y(x− 2) = x+ 1distribute xy− 2y = x+ 1
collect x terms xy− x = 2y+ 1factor x(y− 1) = 2y+ 1
divide x =2y+ 1y− 1
How did you get that?
start y =x+ 1x− 2
cross-mul ply y(x− 2) = x+ 1distribute xy− 2y = x+ 1
collect x terms xy− x = 2y+ 1factor x(y− 1) = 2y+ 1
divide x =2y+ 1y− 1
No-no’s for expressionsI Cannot have zero in thedenominator of anexpression
I Cannot have a nega venumber under an evenroot (e.g., square root)
I Cannot have thelogarithm of a nega venumber
Piecewise-defined functionsExample
Let
f(x) =
{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.
Find the domain and range of fand graph the func on.
Solu onThe domain is [0, 2]. The graphcan be drawn piecewise.
...0..
1..
2..
1
..
2
The range is [0, 2).
Piecewise-defined functionsExample
Let
f(x) =
{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.
Find the domain and range of fand graph the func on.
Solu onThe domain is [0, 2]. The graphcan be drawn piecewise.
...0..
1..
2..
1
..
2
The range is [0, 2).
Piecewise-defined functionsExample
Let
f(x) =
{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.
Find the domain and range of fand graph the func on.
Solu onThe domain is [0, 2]. The graphcan be drawn piecewise.
...0..
1..
2..
1
..
2
.
The range is [0, 2).
Piecewise-defined functionsExample
Let
f(x) =
{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.
Find the domain and range of fand graph the func on.
Solu onThe domain is [0, 2]. The graphcan be drawn piecewise.
...0..
1..
2..
1
..
2
...
The range is [0, 2).
Piecewise-defined functionsExample
Let
f(x) =
{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.
Find the domain and range of fand graph the func on.
Solu onThe domain is [0, 2]. The graphcan be drawn piecewise.
...0..
1..
2..
1
..
2
...
The range is [0, 2).
Functions described numerically
We can just describe a func on by a table of values, or a diagram.
Functions defined by tables I
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 5, 6}.
Functions defined by tables I
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 5, 6}.
Functions defined by tables I
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 5, 6}.
Functions defined by tables I
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 5, 6}.
Functions defined by tables I
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 5, 6}.
Functions defined by tables I
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 5, 6}.
Functions defined by tables II
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 43 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 6}.
Functions defined by tables II
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 43 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 6}.
Functions defined by tables II
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 43 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 6}.
Functions defined by tables II
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 43 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 6}.
Functions defined by tables II
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 43 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 6}.
Functions defined by tables II
Example
Is this a func on? If so, what isthe range?
x f(x)1 42 43 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
Yes, the range is {4, 6}.
Functions defined by tables III
Example
Is this a func on? If so, what isthe range?
x f(x)1 41 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
This is not a func on.
Functions defined by tables III
Example
Is this a func on? If so, what isthe range?
x f(x)1 41 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
This is not a func on.
Functions defined by tables III
Example
Is this a func on? If so, what isthe range?
x f(x)1 41 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
This is not a func on.
Functions defined by tables III
Example
Is this a func on? If so, what isthe range?
x f(x)1 41 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
This is not a func on.
Functions defined by tables III
Example
Is this a func on? If so, what isthe range?
x f(x)1 41 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
This is not a func on.
Functions defined by tables III
Example
Is this a func on? If so, what isthe range?
x f(x)1 41 53 6
Solu on
.....1 ..
2
..
3
.. 4..
5
..
6
This is not a func on.
An ideal function
I Domain is the bu onsI Range is the kinds of sodathat come out
I You can press more thanone bu on to get somebrands
I But each bu on will onlygive one brand
An ideal function
I Domain is the bu ons
I Range is the kinds of sodathat come out
I You can press more thanone bu on to get somebrands
I But each bu on will onlygive one brand
An ideal function
I Domain is the bu onsI Range is the kinds of sodathat come out
I You can press more thanone bu on to get somebrands
I But each bu on will onlygive one brand
An ideal function
I Domain is the bu onsI Range is the kinds of sodathat come out
I You can press more thanone bu on to get somebrands
I But each bu on will onlygive one brand
An ideal function
I Domain is the bu onsI Range is the kinds of sodathat come out
I You can press more thanone bu on to get somebrands
I But each bu on will onlygive one brand
Why numerical functions matterQues on
Why use numerical func ons at all? Formula func ons are so mucheasier to work with.
Answer
I In science, func ons are o en defined by data.I Or, we observe data and assume that it’s close to some nicecon nuous func on.
Why numerical functions matterQues on
Why use numerical func ons at all? Formula func ons are so mucheasier to work with.
Answer
I In science, func ons are o en defined by data.I Or, we observe data and assume that it’s close to some nicecon nuous func on.
Numerical Function ExampleExample
Here is the temperature in Boise, Idaho measured in 15-minuteintervals over the period August 22–29, 2008.
...8/22..
8/23..
8/24..
8/25..
8/26..
8/27..
8/28..
8/29..10 ..
20..
30..
40
..
50
..
60
..
70
..
80
..
90
..
100
Functions described graphicallySome mes all we have is the “picture” of a func on, by which wemean, its graph.
. .
The graph on the right represents a rela on but not a func on.
Functions described graphicallySome mes all we have is the “picture” of a func on, by which wemean, its graph.
.
.
The graph on the right represents a rela on but not a func on.
Functions described graphicallySome mes all we have is the “picture” of a func on, by which wemean, its graph.
. .
The graph on the right represents a rela on but not a func on.
Functions described graphicallySome mes all we have is the “picture” of a func on, by which wemean, its graph.
. .
The graph on the right represents a rela on but not a func on.
Functions described graphicallySome mes all we have is the “picture” of a func on, by which wemean, its graph.
. .
The graph on the right represents a rela on but not a func on.
Functions described verbally
O en mes our func ons come out of nature and have verbaldescrip ons:
I The temperature T(t) in this room at me t.
I The eleva on h(θ) of the point on the equator at longitude θ.I The u lity u(x) I derive by consuming x burritos.
Functions described verbally
O en mes our func ons come out of nature and have verbaldescrip ons:
I The temperature T(t) in this room at me t.I The eleva on h(θ) of the point on the equator at longitude θ.
I The u lity u(x) I derive by consuming x burritos.
Functions described verbally
O en mes our func ons come out of nature and have verbaldescrip ons:
I The temperature T(t) in this room at me t.I The eleva on h(θ) of the point on the equator at longitude θ.I The u lity u(x) I derive by consuming x burritos.
OutlineModeling
Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally
Proper es of func onsMonotonicitySymmetry
MonotonicityExample
Let P(x) be theprobability thatmy income wasat least $x lastyear. Whatmight a graph ofP(x) look like?
Solu on
...
1
..
0.5
..$0..
$52,115..
$100K
MonotonicityExample
Let P(x) be theprobability thatmy income wasat least $x lastyear. Whatmight a graph ofP(x) look like?
Solu on
...
1
..
0.5
..$0..
$52,115..
$100K
Monotonicity
Defini on
I A func on f is decreasing if f(x1) > f(x2) whenever x1 < x2 forany two points x1 and x2 in the domain of f.
I A func on f is increasing if f(x1) < f(x2) whenever x1 < x2 forany two points x1 and x2 in the domain of f.
ExamplesExample
Going back to the burrito func on, would you call it increasing?
AnswerNot if they are all consumed at once! Strictly speaking, theinsa ability principle in economics means that u li es are alwaysincreasing func ons.
Example
Obviously, the temperature in Boise is neither increasing nordecreasing.
ExamplesExample
Going back to the burrito func on, would you call it increasing?
AnswerNot if they are all consumed at once! Strictly speaking, theinsa ability principle in economics means that u li es are alwaysincreasing func ons.
Example
Obviously, the temperature in Boise is neither increasing nordecreasing.
ExamplesExample
Going back to the burrito func on, would you call it increasing?
AnswerNot if they are all consumed at once! Strictly speaking, theinsa ability principle in economics means that u li es are alwaysincreasing func ons.
Example
Obviously, the temperature in Boise is neither increasing nordecreasing.
SymmetryConsider 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 blackhole.
What might their graphs look like?
Possible Intensity Graph
Example
Let I(x) be the intensityof light x distance froma point. Sketch apossible graph for I(x).
Solu on
..x
.
y = I(x)
Possible Gravity Graph
Example
Let F(x) be thegravita onal force at apoint x distance from ablack hole. Sketch apossible graph for F(x).
Solu on
..x
.
y = F(x)
Definitions
Defini on
I A func on f is called even if f(−x) = f(x) for all x in the domainof f.
I A func on f is called odd if f(−x) = −f(x) for all x in thedomain of f.
Examples
Example
I Even: constants, even powers, cosineI Odd: odd powers, sine, tangentI Neither: exp, log
Summary
I The fundamental unit of inves ga on in calculus is the func on.I Func ons can have many representa ons