- Substitute the given \(x=5\) into the function equation \(y = \sqrt{x - 5} - 3\).
- Simplify the expression inside the radical to find \(\sqrt{0} = 0\).
- Subtract \(3\) from the result to find \(y = -3\).
- The corresponding y-value is \(\boxed{-3}\).
Explanation
- Problem Analysis
The problem asks us to evaluate the y-coordinate of the square root function \(y = \sqrt{x - 5} - 3\) given the x-coordinate \(x = 5\). The square root function is defined for values of \(x\) that make the expression inside the radical non-negative. Here, that requires \(x - 5 \geq 0\), which means \(x \geq 5\). Since our given x-value is \(x = 5\), the function is well-defined.
- Evaluating the Function
To find the corresponding y-value, we substitute \(x = 5\) into the given equation:
\[y = \sqrt{x - 5} - 3\]
Substituting \(5\) for \(x\):
\[y = \sqrt{5 - 5} - 3\]
Now, perform the subtraction inside the radical:
\[y = \sqrt{0} - 3\]
The square root of \(0\) is \(0\), so:
\[y = 0 - 3\]
\[y = -3\]
Thus, the ordered pair is \((5, -3)\).
Examples
In real-world applications, such as tracking the growth of an organism or the distance covered by an object over time, square root functions are frequently used to model non-linear relationships. For example, if the distance \(d\) an object travels is related to the time \(t\) by a function like \(d(t) = 2\sqrt{t-1} + 5\), finding the specific position at a certain time \(t\) helps in predicting movement patterns. This evaluation process is fundamental in physics, engineering, and data science.