## Thomas' Calculus 13th Edition

domain: $\mathbb{R}$ range: $[1,\infty)$
The domain of the function is all values of $x$ for which we have a $y$ value (for which the function exists). Since the function is a polynomial and there are no excluded points from the $X$-axis, then the domain is $\mathbb{R}$ (all real numbers). The range of the function consists of all the $y$ values (function values) that correspond to the points in the domain. That is, all values of $1 + x^2$ through all real numbers $x \in \mathbb{R}$. Now, $x^{2} \ge 0$, and so $1 + x^{2} \geq 1$. Thus, the range is $[1,\infty)$.