elleblythe one year ago Greatest Integer Function: In defining [[1-3x]], As x->0+, 1-3x->1*-, [[1-]]=0, why is the sign of the 1* negative? How will we know if it is negative or positive?

1. elleblythe

Also, in defining [[4-x]], how did we come up with 3 and 2? I'm not asking for the solution, just the definition. No solving is needed for these questions.

2. ganeshie8

[x] is the greatest "integer" less than or equal to x

3. ganeshie8

for example [3.5] = 3

4. ganeshie8

see if you can find [2.88]

5. elleblythe

@ganeshie8 Yeah I understand how to find the greatest integer function but my professor requires us to define the greatest integer function, and I'm just wondering how to know the sign of 1*? Why is it (-)?

6. elleblythe

@ganeshie8 Take a loot at the screenshot I posted for a clearer picture of what I'm trying to say.

7. ganeshie8

Yes, can you answer my earlier question

8. elleblythe

@ganeshie8 2

9. ganeshie8

10. elleblythe

@ganeshie8 -6

11. ganeshie8

great! just making sure :) lets get to the question in attachment

12. ganeshie8

|dw:1435246737205:dw|

13. ganeshie8

$$x\to 1^{+}$$ means you're approaching $$1$$ from the right hand side, so let $$x=1.001$$ maybe

14. ganeshie8

[1.001] = ? [4-1.001] = ?

15. elleblythe

@ganeshie8 1 and 2

16. ganeshie8

thats it!

17. elleblythe

@ganeshie8 So how about the (-)??

18. ganeshie8

|dw:1435247060087:dw|

19. elleblythe

@ganeshie8 Sadly, I can't skip it because my teacher requires us to write it down in our exams. How do we know if the value inside the [[]] is coming from the right or left?

20. elleblythe

@ganeshie8 And how did she come up with 3 and 2 in defining [[4-x]]?

21. ganeshie8

$4-n\ge x\gt 3-n$ it was just a guess, you plugin $$n=3$$ : $4-3\ge x\gt 3-3\\~\\1\ge x\gt 0$ which is not what we want so discard this and plugin $$n=2$$ in next trial

22. ganeshie8

|dw:1435247787651:dw|

23. elleblythe

|dw:1435248013287:dw|

24. elleblythe

@ganeshie8

25. ganeshie8

when you substitute $$x=1^{+}$$ into $$4-x$$, you do get something less than $$3$$, which is expressed mathematically as $$3^{-}$$, yes ?

26. ganeshie8

here is how we interpret -,+ in the top of a number : $$3^{+}$$ : right of $$3$$ (example : 3.001) $$3^{-}$$ : left of $$3$$ (example : 2.999)

27. elleblythe

@ganeshie8 Okay I understand now, thank you very much!

28. ganeshie8

yw!