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![Page 1: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/1.jpg)
At once arbitrary yet specific and particular
1
Life without variables is verbose At once arbitrary yet specific and particular
Functions Imaginary square root of -1
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s ) 𝑡=
𝑥𝑣
(𝑖 )2=−1
![Page 2: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/2.jpg)
Life without variables is verbose
4
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
![Page 3: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/3.jpg)
Life without variables is verbose
5
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
![Page 4: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/4.jpg)
Life without variables is verbose
6
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
![Page 5: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/5.jpg)
Life without variables is verbose
7
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
![Page 6: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/6.jpg)
At once arbitrary yet specific and particular
8
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
t0 1 2 3 4 5 6 7 . . .-2 -1
# of seconds
x. . .0 1 2 3 4 5 6 7-2 -1
# of meters
v0 1 2 3 4 5 6 7 . . .-2 -1
# of meters per second
𝑡=𝑥𝑣
? ? ? ??
? ? ? ??
? ? ? ??
![Page 7: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/7.jpg)
At once arbitrary yet specific and particular
9
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
Example duration
Example distance
Example speed
t0 1 2 3 4 5 6 7 . . .-2 -1
# of seconds
x. . .0 1 2 3 4 5 6 7-2 -1
# of meters
v0 1 2 3 4 5 6 7 . . .-2 -1
# of meters per second
0 1 . . .-1t
0 1 . . .-1x
0 1 . . .-1v
= 𝑡=𝑥𝑣
![Page 8: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/8.jpg)
At once arbitrary yet specific and particular
10
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
t, an arbitrary yet specific and particular example of a duration measured in seconds whose number value is chosen from the highlighted domain below
x, an arbitrary yet specific and particular example of a distance measured in meters whose number value is chosen from the highlighted domain below
v, an arbitrary yet specific and particular example of a speed measured in meters per second whose number value is chosen from the highlighted domain below
𝑡=𝑥𝑣
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
=
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
t
x
v=
Obvious now, but easy to forget when doing “calculus of variations,” (i.e. optimization problems)
?? ?
? ??
? ??
![Page 9: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/9.jpg)
At once arbitrary yet specific and particular
11
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
t, an arbitrary yet specific and particular example of a duration measured in seconds whose number value is chosen from the highlighted domain below
x, an arbitrary yet specific and particular example of a distance measured in meters whose number value is chosen from the highlighted domain below
v, an arbitrary yet specific and particular example of a speed measured in meters per second whose number value is chosen from the highlighted domain below
0 1 . . .-1
0 1 . . .-1
=
0 1 . . .-1
𝑡=𝑥𝑣
Obvious now, but easy to forget when doing “calculus of variations,” (i.e. optimization problems)
![Page 10: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/10.jpg)
At once arbitrary yet specific and particular
12
Life without variables is verbose At once arbitrary yet specific and particular
Functions Imaginary square root of -1
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s ) 𝑡=
𝑥𝑣
(𝑖 )2=−1
![Page 11: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/11.jpg)
an ordered pair
Functions
13
an arbitrary yet specific and particular object from collection X
the resulting object in collection Y
The function f
Domain X Codomain YGraph F
Essential stipulation: Each maps to precisely one .
Range of f
![Page 12: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/12.jpg)
14
(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
The “squaring” function f
Domain X
Codomain Y
0 1 2 3 4 . . .-2 -1. . . -4 -3
0 1 2 3 4 . . .-2 -1. . . -4 -3
(0 ,02=0 )(1 ,12=1 )(2 ,22=4 )(−2 , (−2 )2=4 ) 𝑥
0 1 2-2 -1
𝑓 (𝑥 )
1
2
3
4
Graph F
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )=𝑥2Association rule
Functions
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(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
( 𝑦 ,𝑔 (𝑦 ) )𝑦 𝑧=𝑔 ( 𝑦 )
The function gCodomain ZDomain Y Graph G
Composition of functions
15
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(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
( 𝑦 ,𝑔 (𝑦 ) )𝑦 𝑧=𝑔 ( 𝑦 )
The function gCodomain ZDomain Y Graph G
Composition of functions
16
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
( 𝑦 ,𝑔 (𝑦 ) )𝑦 𝑧=𝑔 ( 𝑦 )
The function gCodomain ZDomain Y Graph G
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
![Page 15: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/15.jpg)
𝑥0 1 2-2 -1
𝑔 ( 𝑓 (𝑥 ) )
1
2
3
4
5
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
Composition of functions
17
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
Domain X
Co/domain Y
0 1 2 3 4 5-2 -1-5-4 -3
0 1 2 3 4 5-2 -1-5-4 -3
𝑓 (𝑥 )=𝑥2Graph F
0 1 2 3 4 5-2 -1-5-4 -3Codomain Z
𝑔 ( 𝑦 )=𝑦+1Graph G
![Page 16: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/16.jpg)
Inverses of functions
18
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
![Page 17: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/17.jpg)
𝑥2 3 40 1
𝑔 ( 𝑓 (𝑥 ) )
1
2
3
4
5
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
Inverses of functions
19
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
somethingGraph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
undo somethingGraph G
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
![Page 18: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/18.jpg)
Inverses of functions
20
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
somethingGraph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
undo somethingGraph G
𝑓 (𝑥 )2 3 40 1
5
𝑔 ( 𝑓 (𝑥 ) )1234 STOP
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
𝑥2 3 40 1
5
𝑓 (𝑥 )1234
![Page 19: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/19.jpg)
At once arbitrary yet specific and particular
21
Life without variables is verbose At once arbitrary yet specific and particular
Functions Imaginary square root of -1
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s ) 𝑡=
𝑥𝑣
(𝑖 )2=−1
![Page 20: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/20.jpg)
or
1
2
3
4
or 0 1 2-2 -1 3 4
Square-root “function” and
22
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
𝑓 (𝑥 )=𝑥2Graph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
𝑔 ( 𝑦 )=undosquaring (𝑦 )Graph G
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
Can’t tell which one value to return
? ?
![Page 21: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/21.jpg)
Square-root “function” and
23
𝑦1 2 3 4
-2
-1
𝑔 ( 𝑦 )
1
2
-1-2 0
![Page 22: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/22.jpg)
Square-root “function” and
24
𝑦1 2 3 4
𝑔 ( 𝑦 )
-1-2
-2
-1
1
2
0
![Page 23: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square.](https://reader038.fdocuments.us/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/23.jpg)
1.41 𝑖1.41 𝑖1.41 𝑖
1.411.41
11
000-1
Square-root “function” and
25
𝑦3 4
ℜ [𝑔 ( 𝑦 ) ]
2
-2
-2 0 1 2𝑖
(ℜ [𝑔 (𝑦 ) ]+ 𝑖ℑ [𝑔 (𝑦 ) ] )2
(𝑖 )2=−1
𝑖𝑖 ∙𝑖=−1𝑖𝑖 0 ∙0=0
11 ∙1=1 1.41
1.41 ∙1.41≅ 2
-2
-1
1
2
0
(1.41 𝑖 ) ∙ (1.41 𝑖 )
(1.41 ∙1.41 ) ∙ (𝑖 ∙ 𝑖 )≅−2
𝑔 ( 𝑦 )
𝑖ℑ [𝑔 (𝑦 ) ]