At once arbitrary yet specific and particular

1. At once arbitrary yet specific and particular Life without variables is verbose At once arbitrary yet specific and particular Functions Imaginary square root of -1

2. Life without variables is verbose 5 North 5 North 5 North 4 N 4 N 4 N 3 N 3 N 3 N 2 N 2 N 2 N 1 N 1 N 1 N Main street Main street Main street 1 S 1 S 1 S 2 S 2 S 2 S 3 S 3 S 3 S STOP STOP STOP 4 S 4 S 4 S 5 South 5 South 5 South

3. Life without variables is verbose Example duration 5 North 4 N 3 N 2 N Example distance 1 N Main street 1 S Example speed 2 S 3 S STOP 4 S 5 South

4. Life without variables is verbose Example duration 5 North 4 N 3 N 2 N Example distance 1 N Main street 1 S Example speed 2 S 3 S STOP 4 S 5 South

5. Life without variables is verbose Example duration 5 North 4 N 3 N 2 N Example distance 1 N Main street 1 S Example speed 2 S 3 S STOP 4 S 5 South

6. At once arbitrary yet specific and particular ? ? ? ? ? Example duration 5 North t 4 N . . . -2 -1 0 1 2 3 4 5 6 7 3 N # of seconds ? ? ? ? ? 2 N Example distance 1 N x . . . Main street -2 -1 0 1 2 3 4 5 6 7 # of meters 1 S ? ? ? ? ? Example speed 2 S v 3 S . . . -2 -1 0 1 2 3 4 5 6 7 STOP 4 S # of meters per second 5 South

7. At once arbitrary yet specific and particular x = t v Example duration 5 North t 4 N . . . . . . . . . . . . -2 -1 -1 -1 -1 0 0 0 0 1 1 1 1 2 3 4 5 6 7 3 N # of seconds 2 N Example distance 1 N x . . . Main street -2 -1 0 1 2 3 4 5 6 7 # of meters 1 S Example speed 2 S v 3 S . . . -2 -1 0 1 2 3 4 5 6 7 STOP 4 S # of meters per second 5 South

8. At once arbitrary yet specific and particular Obvious now, but easy to forget when doing “calculus of variations,” (i.e. optimization problems) x = t v x, an arbitrary yet specific and particular example of a distance measured in meters whose number value is chosen from the highlighted domain below 5 North 4 N . . . . . . . . . . . . . . . . . . -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 1 1 1 1 1 t, an arbitrary yet specific and particular example of a duration measured in seconds whose number value is chosen from the highlighted domain below 3 N ? ? ? 2 N = 1 N Main street 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 1 S ? ? ? 2 S 3 S STOP ? 4 S ? ? 5 South

9. At once arbitrary yet specific and particular Obvious now, but easy to forget when doing “calculus of variations,” (i.e. optimization problems) x, an arbitrary yet specific and particular example of a distance measured in meters whose number value is chosen from the highlighted domain below 5 North 4 N . . . . . . . . . -1 -1 -1 0 0 0 1 1 1 t, an arbitrary yet specific and particular example of a duration measured in seconds whose number value is chosen from the highlighted domain below 3 N 2 N = 1 N Main street 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 1 S 2 S 3 S STOP 4 S 5 South

10. At once arbitrary yet specific and particular Life without variables is verbose At once arbitrary yet specific and particular Functions Imaginary square root of -1

11. Functions The functionf Essential stipulation: Each maps toprecisely one. Range of f Domain X Graph F Codomain Y theresulting object in collection Y an ordered pair an arbitrary yet specific and particular object from collection X

12. Functions The “squaring” functionf The functionf Domain X Domain X Graph F Codomain Y Graph F -2 -1 0 1 2 Association rule 4 3 2 1 . . . . . . -4 -4 -3 -3 Codomain Y . . . . . . -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4

13. Composition of functions The functionf The functiong Domain X Graph F Codomain Y Domain Y Graph G Codomain Z

14. Composition of functions The functiong The functionf The functionf The functiong Domain X Graph F Codomain Y Domain Y Graph G Codomain Z Domain Y Graph G Codomain Z Domain X Graph F Codomain Y The functiongf Graph GF Graph G Codomain Z Domain X Graph F Co/domain Y

15. Composition of functions The functiongf -2 -1 0 1 2 Domain X 4 3 Graph GF Graph F 5 Graph G Codomain Z Domain X Graph F Co/domain Y 2 1 Co/domain Y -5 -5 -4 -4 -4 -3 -3 -3 -5 Graph G 5 5 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 Codomain Z

16. Inverses of functions The functiongf Graph GF Graph G Codomain Z Domain X Graph F Co/domain Y

17. Inverses of functions The functiongf Codomain X Domain X 0 1 2 3 4 Domain X 4 3 Graph GF Graph F something 5 Graph G Graph F Co/domain Y 2 1 Co/domain Y . . . . . . . . . -4 -4 -4 -3 -3 -3 Graph G undo something . . . . . . . . . -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 Codomain X

18. Inverses of functions The functiongf 0 0 1 1 2 2 3 3 4 4 4 4 Domain X 3 3 5 5 Graph GF Graph F something 2 2 Graph G Codomain X Domain X Graph F Co/domain Y 1 1 Co/domain Y . . . . . . . . . -4 -4 -4 -3 -3 -3 Graph G undo something . . . . . . . . . -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 STOP Codomain X

19. At once arbitrary yet specific and particular Life without variables is verbose At once arbitrary yet specific and particular Functions Imaginary square root of -1

20. Square-root “function” and The functiongf or -2 -1 0 1 2 Domain X 4 3 Graph GF Graph F Graph G Codomain X Domain X Graph F Co/domain Y 2 1 Co/domain Y or . . . . . . . . . -4 -4 -4 -3 -3 -3 Graph G . . . . . . . . . -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 ? ? 3 4 Codomain X Can’t tell which one value to return

21. Square-root “function” and 1 2 0 -1 3 -2 4 -1 2 1 -2

22. Square-root “function” and 1 2 0 -1 3 -2 4 -1 2 1 -2

23. Square-root “function” and -2 0 0 -1 3 -2 1 2 4 -1 2 2 1 -2