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
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..
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!
![Page 2: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/2.jpg)
Section 1.1Functions and theirRepresentations
V63.0121.001, Calculus IProfessor Ma hew Leingang
New York University
![Page 3: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/3.jpg)
Announcements
I First WebAssign-mentsare due January 31
I Do the Get-to-Know-Yousurvey for extra credit!
![Page 4: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/4.jpg)
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.
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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.
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OutlineModeling
Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally
Proper es of func onsMonotonicitySymmetry
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The Modeling Process
...Real-worldProblems
.. Mathema calModel
.
.
Mathema calConclusions
.
.
Real-worldPredic ons
. model.
solve
.
interpret
.
test
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Plato’s Cave
..
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The Modeling Process
...Real-worldProblems
.. Mathema calModel
.
.
Mathema calConclusions
.
.
Real-worldPredic ons
. model.
solve
.
interpret
.
test
.Sh
adow
s.Form
s
![Page 10: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/10.jpg)
OutlineModeling
Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally
Proper es of func onsMonotonicitySymmetry
![Page 11: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/11.jpg)
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.
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Formula function exampleExample
Let f(x) =x+ 1x− 2
. Find the domain and range of f.
Solu on
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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 }
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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 }
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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
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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
![Page 17: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/17.jpg)
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
![Page 18: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/18.jpg)
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
![Page 19: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/19.jpg)
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
![Page 20: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/20.jpg)
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
![Page 21: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/21.jpg)
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
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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).
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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).
![Page 24: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/24.jpg)
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).
![Page 25: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/25.jpg)
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).
![Page 26: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/26.jpg)
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).
![Page 27: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/27.jpg)
Functions described numerically
We can just describe a func on by a table of values, or a diagram.
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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}.
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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}.
![Page 30: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/30.jpg)
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}.
![Page 31: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/31.jpg)
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}.
![Page 32: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/32.jpg)
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}.
![Page 33: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/33.jpg)
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}.
![Page 34: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/34.jpg)
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}.
![Page 35: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/35.jpg)
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}.
![Page 36: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/36.jpg)
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}.
![Page 37: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/37.jpg)
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}.
![Page 38: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/38.jpg)
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}.
![Page 39: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/39.jpg)
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}.
![Page 40: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/40.jpg)
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.
![Page 41: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/41.jpg)
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.
![Page 42: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/42.jpg)
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.
![Page 43: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/43.jpg)
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.
![Page 44: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/44.jpg)
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.
![Page 45: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/45.jpg)
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.
![Page 46: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/46.jpg)
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
![Page 47: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/47.jpg)
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
![Page 48: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/48.jpg)
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
![Page 49: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/49.jpg)
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
![Page 50: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/50.jpg)
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
![Page 51: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/51.jpg)
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.
![Page 52: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/52.jpg)
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.
![Page 53: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/53.jpg)
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
![Page 54: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/54.jpg)
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.
![Page 55: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/55.jpg)
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.
![Page 56: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/56.jpg)
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.
![Page 57: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/57.jpg)
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.
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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.
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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.
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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.
![Page 61: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/61.jpg)
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.
![Page 62: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/62.jpg)
OutlineModeling
Examples of func onsFunc ons expressed by formulasFunc ons described numericallyFunc ons described graphicallyFunc ons described verbally
Proper es of func onsMonotonicitySymmetry
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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
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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
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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.
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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.
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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.
![Page 68: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/68.jpg)
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.
![Page 69: Lesson 1: Functions and their representations (slides)](https://reader035.vdocument.in/reader035/viewer/2022081403/556150b3d8b42a8a7d8b4f59/html5/thumbnails/69.jpg)
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?
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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)
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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)
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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.
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Examples
Example
I Even: constants, even powers, cosineI Odd: odd powers, sine, tangentI Neither: exp, log
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Summary
I The fundamental unit of inves ga on in calculus is the func on.I Func ons can have many representa ons