What is the smallest x-value in this function?
$\begin{array}{c}
y=\sqrt{x-5}-3 \\
([?],
\end{array}$
To determine the smallest \(x\)-value in the domain of the function \(y = \sqrt{x-5} - 3\), we need to consider the definition of the square root function for real numbers. For the expression \(\sqrt{x-5}\) to yield a real value, the radicand (the part inside the square root) must be non-negative. This is because the square root of a negative number is not a real number. Therefore, we establish the following condition:
\[x - 5 \geq 0\]
To find the values of \(x\) for which the function is defined, we solve the inequality derived in the previous step:
\[x - 5 \geq 0\]
Adding 5 to both sides of the inequality, we get:
\[x \geq 5\]
This inequality describes the entire domain of the function, which is the interval \([5, \infty)\). The smallest value in this interval, and thus the smallest \(x\)-value for which the function is defined, is the boundary value of the inequality.
Since the function \(y = \sqrt{x-5} - 3\) is defined for all \(x \geq 5\), the smallest possible \(x\)-value is 5. At \(x = 5\), the function value is \(y = \sqrt{5-5} - 3 = 0 - 3 = -3\). Thus, the starting point of the graph of this function is at \((5, -3)\). Therefore, the smallest \(x\)-value is \(\boxed{5}\).
Understanding the domain of a square root function is essential in various fields, such as physics, when analyzing the motion of an object. For instance, if you are modeling the time-dependent position or velocity of an object using a square root function, the domain tells you exactly when the physical situation becomes valid. In our example, if \(x\) represents time, the function only models the scenario for times starting at \(x = 5\) units. Knowing this boundary helps prevent errors when interpreting data or predicting future outcomes based on mathematical models.