What is the domain of the function 1/(x^2+x+1)+x ?

Apr 2, 2017

$\mathbb{R}$

Explanation:

Notice that:

${x}^{2} + x + 1 = {\left(x + \frac{1}{2}\right)}^{2} + \frac{3}{4}$

So:

${x}^{2} + x + 1 \ge \frac{3}{4} > 0$ for all real values of $x$

Hence, for any real value of $x$, the given expression:

$\frac{1}{{x}^{2} + x + 1} + x$

is well defined.

So the domain is the whole of the real numbers $\mathbb{R}$.