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I have $f(x)$ and need to find the intervals when it is is increasing and when it is deacreasing. I found out that it is not defined at $x=-1$ and $x=1$, and it made me unsecure. When I write the intervals, do I need to exclude these points, or not because they are not in the domain? And do they mean strictly increasing or just monotonically increasing?

Is it correct to say that it is increasing on $[-sqrt(3),sqrt(3)]\{-1,1}]$ and decreasing on $(-inf,-sqrt(3)] U [sqrt(3),inf)$?

netwon1227
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    If you use the derivative, you will find the points at which the derivative is $0$. Add to these the points $-1$ and $1$. That will give you all the points where the original function could (it need not) change from increasing to decreasing, or vice-versa. Check all these points. Life is made easier by the fact that $f(-x)=-f(x)$, so there is a kind of symmetry, the part with $x\lt 0$ is obtained by rotating the part with $x\gt 0$ by $180$ degrees about the origin. – André Nicolas Oct 25 '15 at 16:52
  • Yes, so do I need to say that it is increasing on (-sqrt(3),-1)U(-1,0)U(0,1)U(1,sqrt(3))? – netwon1227 Oct 25 '15 at 17:05

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When you take the derivative you should find that the derivative is not defined at -1 and 1.

For your second question: Strictly increasing functions have $f(x)>f(y)$ implies $x>y$ and vice versa, and $f(x)=f(y)$ means $x=y$. Monotonic functions do not have this second part, so it could be constant on an interval then continue to increase after.

Faraz Masroor
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