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