Math Is Fun Forum

pineapple12

Member

Registered: 2016-06-21

Posts: 8

Algebra Functions

Define {x} = x- \lfloor x\rfloor. That is to say, {x} is the "fractional part" of x. If you were to expand the number x as a decimal, {x} is the stuff after the decimal point. For example {3/2} = 0.5 and {\pi} = 0.14159...

Now, using the above definition, determine if the function below is increasing, decreasing, even, odd, and/or invertible on its natural domain:

f(x) = \lfloor x \rfloor - {x}

I'm kind of stuck on this one as I usually use a graph to help me determine if the function is increasing, decreasing, even, odd, etc. (oops, I know it's not a very reliable way) and for this one I have no idea what the graph would look like because of that floor function. How should I do this problem?

Any help would be appreciated! Thanks!

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,444

Website

Re: Algebra Functions

Plug in some numbers x belonging to the interval [0,1] into f. Then do the same for [1,2], and [-1,0], [-2,-1]. What is happening here?

This plot of the function might give you some idea of what is going on.

Welcome to the forum!

Last edited by zetafunc (2016-06-23 01:46:11)

pineapple12

Member

Registered: 2016-06-21

Posts: 8

Re: Algebra Functions

I see that for every interval, as x increases, the graph decreases. And, at every whole number, the graph just starts over and decreases again. Thanks so much for the graph!!!

Ok, so I think the graph is neither even or odd, and not invertible either. I'm not sure if it's increasing or decreasing though....

I cannot thank you enough! Thanks!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Functions

x	f(x)
1	1
1.001	0.999
1.4	0.6
1.6	0.4
1.8	0.2
2	2
2.2	1.8
2.4	1.6
2.6	1.4
2.8	1.2
3	3
3.2	2.8
3.4	2.6
3.6	2.4
3.9999	2.0001
4	4

this is the type of variation of the function between x=1 to x=4

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

pineapple12

Member

Registered: 2016-06-21

Posts: 8

Re: Algebra Functions

So, is it decreasing??? There's not a line in the graph that increases although I'm not sure.

Also, I was wrong about the graph being neither even or odd, and not invertible either. Not sure what I did wrong.

Sorry...

Thanks anyway

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Functions

the graph has discontinuity when x crosses a whole number. f9x) increases by 2. between a whole number and the next graph is decreasing.globally it is making higher tops and higher bottoms. i do not howit is treated in mathematics. Is it oscillatory or something else?

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}