Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

I used

FunctionDomain[1/(x^3 + 1), x, Complexes]

and got

1 + x^3 != 0

Is this a bug in FunctionDomain (V.10)?

Can I receive $x \neq -1$, $x \neq \dfrac{1-\sqrt{3}I}{2}$ and $x \neq \dfrac{1+\sqrt{3}I}{2}$? With Maple, I tried enter image description here

If I used

FunctionDomain[1/(x^3 + 1), x]

I got true answer

x < -1 || x > -1

share|improve this question
    
I don't think your Maple comparison is quite fair. You've explicitly asked Maple to solve an inequality, rather than asked for the domain of a function. That doesn't account for more complicated examples, as I've placed in my answer. –  Mark McClure Jul 16 at 16:52

1 Answer 1

up vote 12 down vote accepted

This is not a bug; it is an implicit representation of the domain. Note that condition $1-x^3 \neq 0$ cannot be simplified further over the complexes or, at least, Reduce doesn't simplify it further:

Reduce[1 - x^3 != 0, x, Reals] 
(* Out: x < 1 || x > 1 *)

Reduce[1 - x^3 != 0, x, Complexes] 
(* Out: -1 + x^3 != 0 *)

A similar thing can happen over the reals:

FunctionDomain[1/(2 + Sin[x] + Cos[Pi*x]), x]
(* Out: Cos[Pi*x] + Sin[x] != -2 *)

The expression $\cos(\pi x) + \sin(x)$ is never equal to $-2$, by the way, but Reduce can't prove that and I don't think it's reasonable to expect it to do so. Is this a bug? Probably not; the answer is correct.

Here's another example:

FunctionDomain[1/Sqrt[Sin[1/x] - x], x]
(* Out: x != 0 && x - Sin[1/x] < 0 *)

How might we improve this result? The graph of $x-\sin(1/x)$ crosses the $x$-axis infinitely many times near $x=0$ and at values that are not expressible in closed form.

Plot[Sin[1/x] - x, {x, 0, 1/2},
  PlotPoints -> 200, MaxRecursion -> 12]

enter image description here

Considering these types of examples, I think that implicit representations are quite reasonable - even preferable. In simple cases, it's quite easy to extract out Unequal expressions, solve the corresponding equations, and come to the conclusion that you did with Maple.

share|improve this answer
    
Mclure First, thanks to your answer. I think, we want to find the domain of the given function, we must solve the inequality x^3+1<>0. –  minthao_2011 Jul 17 at 2:01

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.