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

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

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

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

? ? ? ??

? ? ? ??

? ? ? ??

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

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)

?? ?

? ??

? ??

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)

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

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

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

(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

(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

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

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

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

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

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

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

? ?

Square-root “function” and

23

𝑦1 2 3 4

-2

-1

𝑔 ( 𝑦 )

1

2

-1-2 0

Square-root “function” and

24

𝑦1 2 3 4

𝑔 ( 𝑦 )

-1-2

-2

-1

1

2

0

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 ) ]