Find the domain of $y=\sqrt{3 x+3}$.
To determine the domain of the function \(y = \sqrt{3x + 3}\), we must ensure that the expression inside the square root (the radicand) is non-negative, as the square root of a negative number is not a real number. Therefore, we set up the following inequality:
\[3x + 3 \geq 0\]
Now, we solve the inequality \(3x + 3 \geq 0\) for \(x\). First, subtract \(3\) from both sides:
\[3x \geq -3\]
Next, divide both sides by \(3\):
\[x \geq -1\]
This inequality defines the set of all real numbers \(x\) for which the function is defined.
Comparing our result \(x \geq -1\) with the provided options:
The correct option is \(x \geq -1\). Thus, the domain of the function is all \(x\) such that \(x \geq -1\). Therefore, the final answer is \(\boxed{x \geq -1}\).
In many real-world scenarios, certain mathematical models have constraints. For example, if you are designing a budget for a project where the cost is modeled by a square root function \(C(t) = \sqrt{3t + 3}\) (in thousands of dollars), where \(t\) represents time in months from a starting point, you must ensure the input is valid. Since the cost cannot involve the square root of a negative number in this context, the domain \(t \geq -1\) informs you that this specific model is only applicable for time periods starting from one month before the project launch, ensuring the financial planning remains mathematically sound and realistic.