Find the domain of $y=\sqrt{x+6}$.
The objective is to find the domain of the function \(y = \sqrt{x+6}\). The domain of a function consists of all possible input values (\(x\)-values) for which the function is defined. In the case of a square root function of the form \(y = \sqrt{u}\), the expression \(u\) inside the root must be greater than or equal to zero, because the square root of a negative number is not a real number. Therefore, we must satisfy the condition \(u \geq 0\).
In the given function \(y = \sqrt{x+6}\), the expression inside the square root is \(u = x + 6\). To ensure that \(y\) is a real number, we set the condition:
\[x + 6 \geq 0\]
To solve for \(x\), we subtract \(6\) from both sides of the inequality:
\[x \geq -6\]
This result means that the function \(y = \sqrt{x+6}\) is defined for all values of \(x\) greater than or equal to \(-6\).
We have determined that the domain is \(x \geq -6\). Comparing this result with the given choices:
The correct option is \(x \geq -6\).
In real-world applications, such as structural engineering, the square root function often appears when calculating the length of a support beam based on the dimensions of a base. If the design formula is related to \(\sqrt{x+6}\), where \(x\) represents a variable offset, knowing the domain allows engineers to determine the permissible range for that offset, ensuring that the physical dimensions remain real and meaningful. Recognizing domains is essential for verifying that models do not break under negative input parameters.