Intuitive explanation on the function

Question

$$h(x) = x^2 + 2x^{-1}$$

a)Identify the natural domain of $h$, asymptotes and the interval(s) on which $h(x)\lt0$

Based on the given function, it is undefined at $x = 0$, so the natural domain would be any number except $0$. And based on the function, the function has a vertical asymptote ie at $x=0$, which is basically the $y$ axis.

I'm a bit confused with the last question although the only rational way would be to set the equation to be $\le0$ and try to solve for $x$ which didn't give me a precise interval. Could someone explain intuitively or arithmetically as to how to go about it?

• Is it $<$ or $\le$ ? – Yves Daoust Feb 16 '17 at 10:43
• What do you mean it doesn't give you a precise interval? – Jack M Feb 16 '17 at 10:44
• Sorry, it's <0. I mean computing that would give me the cube root of negative 2 which would be approximately -1.26, so would x<-1.26 be my interval for the same? – Gary Andrews30 Feb 16 '17 at 10:45
• $(-\sqrt2, 0)$ sounds like a pretty precise interval to me. – Jack M Feb 16 '17 at 12:34

You are asked to solve the inequation

$$x^2+2x^{-1}<0.$$

Assuming $x>0$, you can multiply by $x$ and get

$$x^3+2<0,$$ which is not possible.

Then assuming $x<0$,

$$x^3+2>0,\\x^3>-2,\\ x>-\sqrt2.$$

Finally,

$$-\sqrt2<x<0.$$

This curve is known as the "Trident of Newton", or the "Parabola of Descartes". http://www.mathcurve.com/courbes2d/trident/trident.shtml

• +1 For the additional information about the name of the function:). – MrYouMath Feb 16 '17 at 11:17
• @MrYouMath: the site linked to is a sheer wonder. – Yves Daoust Feb 16 '17 at 11:19
• Already bookmarked the site ;-). – MrYouMath Feb 16 '17 at 11:21