What is the domain of this square root function?
$\begin{array}{c}
f(x)=\sqrt{6 x-24}+5 \\
x \geq[?]
\end{array}$
To determine the domain of the function \(f(x) = \sqrt{6x - 24} + 5\), we must identify the values of \(x\) for which the function is defined. A square root function \(\sqrt{u}\) is only defined when the radicand \(u\) is a real, non-negative number. Therefore, we require the expression inside the square root to be greater than or equal to zero.
We set the radicand \(6x - 24\) to be non-negative:
\[6x - 24 \geq 0\]
To solve for \(x\), we first add \(24\) to both sides of the inequality:
\[6x \geq 24\]
Next, we divide both sides by \(6\):
\[x \geq \frac{24}{6}\]
\[x \geq 4\]
Thus, the function \(f(x) = \sqrt{6x - 24} + 5\) is defined for all \(x \geq 4\).
The domain of the function is the set of all real numbers \(x\) such that \(x \geq 4\). Comparing this to the required form \(x \geq [?]\), we find that the value is \(4\). Therefore, the final result is \(\boxed{4}\).
Imagine you are designing a safety feature for a rollercoaster that involves a square root tracking system. The height of a specific segment of the track relative to the base is modeled by the function \(h(x) = \sqrt{6x - 24} + 5\), where \(x\) represents the horizontal distance from the start of the segment. Because the track cannot exist where the height calculation results in an imaginary number, the coaster's sensors must be calibrated to only operate when the horizontal distance is at least 4 meters. Understanding how to calculate the domain ensures that the system is properly constrained, preventing errors in real-time safety monitoring.