Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have plotted function $(1+\frac{1}{x})^{x}$ using Maple and got following graph:

enter image description here

So it seems that function isn't defined on $(-1,0)$ interval , but if I take that $x=\frac{-1}{3}$ I can write:

$$y=(1+\frac{1}{\frac{-1}{3}})^{\frac{-1}{3}}=(1-3)^{\frac{-1}{3}}=(-2)^{\frac{-1}{3}}=\frac{1}{(-2)^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{-2}}=\frac{-1}{\sqrt[3]{2}}$$

which is real number.So my question is :Is function $(1+\frac{1}{x})^{x}$ defined on $(-1,0)$ interval or not ?

share|improve this question

3 Answers 3

up vote 2 down vote accepted

Another way to solve this is to allow complex values, and choose the principal value in the usual way. Then your function IS defined and continuous on $(-1,0)$, but has non-real values everywhere there!

Here it is, real part in green, imaginary part in red:

graph

share|improve this answer
    
so,I can conclude that function is continuous only if codomain is $\mathbb{C}$ ? –  pedja Oct 30 '11 at 12:51
    
That function in my picture is continuous. Don't know about your "only if" though. –  GEdgar Oct 30 '11 at 13:08

I guess here there is kind o a philosophical problem: for instance, if you take $x=-\frac{1}{2}$, you face the problem $\sqrt{-1}$. So one can say that there are some points in $(-1,0)$ where the function is defined and some other where is not. In general, it's quite difficult to say a priori which are the right points and so, in order to avoid contradiction, one prefers to exclude $(-1,0)$ from the domain.

This i by the way also what happens for the function $x^x$. One says that its domain is $(0,\infty)$, even if it is clear that it is well defined also for particular negative real numbers.

share|improve this answer
    
Let's suppose that I have some number of infected persons among healthy ones..does it mean that I should quarantined all of them because I don't know which of them are infected ? –  pedja Oct 30 '11 at 11:40
    
@pedja: Yes. At least When the non-infected people form a set of measure zero. Also when allowing them to escape the quarantine poses a hazard to public safety! –  Jyrki Lahtonen Oct 30 '11 at 12:10
    
@pedja: Yes, Jyrki's reason above shows that it would be impossible for a computer to write down a picture. Also because I don't think there is a closed formula to find the non-infected persons. –  Valerio Capraro Oct 30 '11 at 12:17
    
I will stop thinking about math as an exact science –  pedja Oct 30 '11 at 12:37
1  
He, he... –  J. M. Oct 30 '11 at 12:50

The expression $x^y$ is defined for $x>0$ if $y\not \in Z $ in the case of real numbers. Otherwise we could write, for example, $-\sqrt[3]{2}=\sqrt[3]{-2}=(-2)^{1/3}=(-2)^{2/6}=\sqrt[6]{(-2)^2}=\sqrt[6]{2^2}=\sqrt[3]{2}$. Contradiction.

Sincerely,

Tigran

share|improve this answer

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.