Determine the Domain of √(x²+2)/3

To solve this problem, we'll determine the domain of the function $f(x) = \frac{\sqrt{x^2+2}}{3}$.

First, consider the expression inside the square root, $x^2 + 2$. In order for the square root to be defined for real numbers, the expression $x^2 + 2$ must be non-negative.

Let's analyze $x^2 + 2$:

• For any real number $x$, the expression $x^2$ is always non-negative.
• Adding 2 to $x^2$ means $x^2 + 2$ is always greater than or equal to 2.
• Thus, $x^2 + 2 \geq 2 > 0$ for all real numbers $x$.

Since the value under the square root is always positive for all real numbers, the square root, and hence the function $f(x)$, is defined for all real numbers.

Therefore, the function has no restrictions on its domain other than the real number system itself. There are no variables in the denominator that can make it zero, as it is the constant 3.

Thus, the domain of the function is all real numbers.

The correct answer choice is: All real numbers.