What is the **domain** of the real-valued function $g(x) = x^{3} + \sqrt{x + 2} - 10$?

The domain of a real-valued function is the set of inputs for which the output is a real number. Examine each piece of $g(x) = x^{3} + \sqrt{x + 2} - 10$: - $x^{3}$ is defined for all real $x$. - $\sqrt{x + 2}$ is real **only when** the radicand is non-negative: $x + 2 \ge 0$, i.e. $x \ge -2$. - $-10$ is a constant. The overall function is defined wherever **every** piece is defined. The binding constraint is the square root: $x \ge -2$. - Trap A ignores the square-root restriction. - Trap B treats $\sqrt{x+2}$ as if it were $1/(x+2)$ — i.e. excludes a single point rather than a half-line. - Trap D solves $x - 2 \ge 0$ instead of $x + 2 \ge 0$ (sign error inside the radical). The domain is $\{x \in \mathbb{R} : x \ge -2\}$, or in interval notation $[-2, \infty)$.