Find the domain of $y=\sqrt{x+6}$.
To find the domain of the function \(y = \sqrt{x+6}\), we must identify all real numbers \(x\) for which the expression \(y\) is defined. The square root function, \(\sqrt{u}\), is defined in the real number system if and only if the radicand (the expression inside the root) is greater than or equal to zero. In this case, the expression inside the root is \(x + 6\). Therefore, for the function to yield real values, we must have \(x + 6 \geq 0\).
To solve the inequality \(x + 6 \geq 0\) for \(x\), we subtract \(6\) from both sides of the inequality:
\[x + 6 - 6 \geq 0 - 6\]
\[x \geq -6\]
This result means that the function \(y = \sqrt{x+6}\) is defined for any value of \(x\) that is greater than or equal to \(-6\). This set of numbers constitutes the domain of the function. Comparing this with the given options, we identify that the correct range is \(x \geq -6\).
In fields like engineering or physics, domain restrictions are essential. For example, if a function represents the time \(t\) elapsed during an experiment, its domain might be \(t \geq 0\), as negative time often isn't applicable. In our case, the function \(y = \sqrt{x+6}\) represents a geometric or physical relationship that requires \(x+6\) to be non-negative, helping us model real-world constraints accurately. Understanding domains helps ensure that mathematical models operate within logical and physical boundaries.