Find the domain of $y=4 \sqrt{4 x+2}$.
The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real number. For the square root function \(y = 4\sqrt{4x + 2}\), the square root of a number is only defined in the set of real numbers if the expression inside the square root (the radicand) is greater than or equal to zero. If the radicand were negative, the function would result in an imaginary number, which is not included in the real-valued domain.
Given the function \(y = 4\sqrt{4x + 2}\), we identify the radicand as \(4x + 2\). To ensure the function outputs real values, we must satisfy the inequality:
\[4x + 2 \geq 0\]
This is a linear inequality that we can solve using basic algebraic operations. First, isolate the term containing \(x\) by subtracting 2 from both sides of the inequality:
\[4x \geq -2\]
Next, divide both sides by 4 to solve for \(x\):
\[x \geq \frac{-2}{4}\]
Simplifying the fraction, we get:
\[x \geq -\frac{1}{2}\]
Therefore, the domain of the function is all real numbers \(x\) such that \(x \geq -\frac{1}{2}\).
The solution \(x \geq -\frac{1}{2}\) corresponds to the set of all real numbers starting from \(-\frac{1}{2}\) and extending infinitely to the right. This matches the fourth option provided in the problem. The final domain is \(x \geq -\frac{1}{2}\).
Understanding domains is essential in fields like engineering and physics, where variables represent physical quantities such as time, distance, or temperature. For instance, if a formula describes the speed of a vehicle in terms of the amount of fuel in its tank, the domain ensures you don't calculate speeds for negative fuel levels, which is physically impossible. By identifying the domain, we ensure that our models remain grounded in reality and provide valid, meaningful results for real-world scenarios.