Determine the Domain of: √(3x²+7)/12

Look at the following function:

$\frac{\sqrt{3x^2+7}}{12}$

What is the domain of the function?

To find the domain of the function $\frac{\sqrt{3x^2+7}}{12}$, we must ensure that the function is defined for all real numbers.

Step 1: Evaluate the expression under the square root, $3x^2 + 7$, which must be non-negative. Since it's a quadratic expression in the form of $ax^2 + b$, compute for any potential zero or negative range.

Step 2: Notice that $3x^2 + 7 \geq 0$ for all $x$ because $3x^2 \geq 0$ for any real number $x$ and adding 7 makes this entire expression always positive (i.e., tends upwards away from zero).

Step 3: As the denominator $12$ is a positive constant, it imposes no additional restrictions on the domain. Thus, the function is defined wherever the numerator is defined.

Conclusion: Since there's no $x$ that makes $3x^2 + 7 < 0$, the function is defined for all real numbers.

This means the domain of the function is all real numbers, confirmed by choice number 1: $\text{All real numbers}$.