# How do you find the domain and range for y = sqrt(x^2 + 2x + 3)?

${x}^{2} + 2 x + 3 = {x}^{2} + 2 x + 1 + 2$
$= {\left(x + 1\right)}^{2} + 2 \ge 2 \forall x \in \mathbb{R}$
So $\sqrt{{x}^{2} + 2 x + 3}$ is defined $\forall x \in \mathbb{R}$ and the domain is the whole of $\mathbb{R}$.
Also we see that the range is $\left\{y \in \mathbb{R} : y \ge 2\right\}$. The minimum value $y = 2$ occurs when $\left(x + 1\right) = 0$, i.e. when $x = - 1$.