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At once arbitrary yet specific and particular. Life without variables is verbose. At once arbitrary yet specific and particular. X. F. Y. -2. -1. -1. -1. 0. 0. 0. 1. 1. 1. 0. 3. 1. 2. 4. -1. -2. Functions. Imaginary square root of -1. -1. 2. 2. 1. -2. - PowerPoint PPT Presentation
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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 )
(𝑥 , 𝑓 (𝑥 ) )𝑥𝑦= 𝑓 (𝑥 )
X YF
𝑡=𝑥𝑣
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
(𝑖 )2=−1
-1 3 4
-2
-12
-21
2
-2 0 1 2
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 )
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
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
At once arbitrary yet specific and particular
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
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
? ? ? ??
? ? ? ??
? ? ? ??
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
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
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
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)
?? ?
? ??
? ??
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
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
=
Obvious now, but easy to forget when doing “calculus of variations,” (i.e. optimization problems)𝑡=𝑥
𝑣
At once arbitrary yet specific and particular
11
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
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
(𝑖 )2=−1
-1 3 4
-2
-12
-21
2
-2 0 1 2(𝑥 , 𝑓 (𝑥 ) )𝑥𝑦= 𝑓 (𝑥 )
X YF
an ordered pair
Functions
12
an arbitrary yet specific and particular (asap)
object from collection X
the resulting object in the collection Y
The function f
Domain X Codomain YGraph F
Functions
13
(𝑥 , 𝑓 (𝑥 ) )𝑥𝑦= 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 )
𝑓 (𝑥 )=𝑥2Association rule
𝑥0 1 2-2 -1
𝑓 (𝑥 )
1
2
3
4
Graph F
(𝑥 , 𝑓 (𝑥 ) )
(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
14
(𝑥 , 𝑓 (𝑥 ) )𝑥𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 , 𝑓 (𝑥 ) )𝑥𝑦= 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
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
15
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
f
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
g
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain Z
𝑔 ( 𝑦 )=𝑦+1Graph G
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
16
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
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
g
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
undo somethingGraph G
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
Inverses of functions
17
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
g
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
At once arbitrary yet specific and particular
18
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
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
(𝑖 )2=−1
-1 3 4
-2
-12
-21
2
-2 0 1 2(𝑥 , 𝑓 (𝑥 ) )𝑥𝑦= 𝑓 (𝑥 )
X YF
or 0 1 2-2 -1
or
1
2
3
4
3 4
Square-root “function” and
19
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
f 𝑓 (𝑥 )=𝑥2Graph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
g 𝑔 ( 𝑦 )=√ (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
Square-root “function” and
20
𝑦1 2 3 4
-2
-1
𝑔 ( 𝑦 )
1
2
-1-2 0
Square-root “function” and
21
𝑦1 2 3 4
-2
-1
1
2
𝑔 ( 𝑦 )
-1-2 0
-1
Square-root “function” and
22
𝑦3 4
-2
-1
ℜ [𝑔 ( 𝑦 ) ]
2
-2
𝑖ℑ [𝑔 (𝑦 ) ]
1
2
-2 0 1 2𝑖
(ℜ [𝑔 (𝑦 ) ]+ 𝑖ℑ [𝑔 (𝑦 ) ] )2
(𝑖 )2=−1