What is the domain of this square root function?
$\begin{array}{c}
f(x)=\sqrt{x+9}-21 \\
x \geq[?]
\end{array}$
The function provided is \(f(x) = \sqrt{x+9} - 21\). In the set of real numbers, the square root of a value is defined only if the expression inside the radical (the radicand) is non-negative. This is because the square of any real number is always non-negative. Therefore, for the function \(f(x)\) to be defined, the expression under the square root, which is \(x+9\), must be greater than or equal to zero.
To find the domain, we set up the following inequality:
\[x + 9 \geq 0\]
This inequality represents the condition that the input \(x\) must satisfy so that the square root is defined for real numbers. We subtract 9 from both sides of the inequality to isolate \(x\):
\[x \geq -9\]
Thus, the domain of the function consists of all real numbers \(x\) such that \(x\) is greater than or equal to -9.
Comparing our result \(x \geq -9\) to the format \(x \geq [?]\), we identify that the missing value is -9. This means that for any value less than -9, the term \(\sqrt{x+9}\) results in an imaginary number, so the domain is restricted to the interval \([-9, \infty)\). The final answer is \(\boxed{-9}\).
Understanding domain restrictions is crucial in real-world scenarios such as engineering and physics. For example, if a function calculates the velocity of an object based on the time \(t\) passed since some event, and the formula involves a square root like \(\sqrt{t-5}\), you can only calculate the velocity for \(t \geq 5\). Trying to calculate the velocity at \(t=3\) would lead to a domain error, indicating the model is only valid from the start time at 5 seconds onwards. This helps professionals avoid invalid calculations and understand the operational limits of their models.