Find intervals where f is continuous, $f(x)=\sqrt{x^{2}+x+1}$ Find intervals where f is continuous, $f(x)=\sqrt{x^{2}+x+1}$. I would appreciate any help. I know that $x^2+x+1$ is continuous, but I'm not sure how to deal with the composition of this function with the square root function - and I'm not sure how to deal with the fact that the radicand must be positive for the function to be defined at all.
 A: It seems like you already have a few of the ingredients needed to prove the continuity of this function.
A rather crucial property of continuity is that the composition of two continuous functions is continuous - this is certainly something you should seek to prove if you haven't done so before. So, if you can prove  both that $x^2+x+1$ is continuous and that $\sqrt{x}$ is, so must their composition be.
Now, the other consideration is that the square root function requires a positive input, so we need to figure out where $x^2+x+1$ is positive in case the function strays out of the domain of $\sqrt{\cdot}$. There are various ways to do this - the simplest is to plug it into the quadratic formula, see that its roots are $\frac{-1\pm \sqrt{-3}}{2}$, neither of which is real, implying that it never crosses zero and thus must be positive everywhere. Another somewhat more general way would be to note that $x^2+x+1 \geq 1+x$ (the tangent at $x=1$) and $x^2+x+1 \geq -x$ (a tangent line at $x=-1$), and since one of $1+x$ or $-x$ is positive, $x^2+x+1$ must be as well as it is greater than both.
Putting all this together should allow you to show that $\sqrt{x^2+x+1}$ is a continuous function.
