The range of which function includes -4?
We are evaluating the range of four functions: \(y = \sqrt{x} - 5\), \(y = \sqrt{x} + 5\), \(y = \sqrt{x + 5}\), and \(y = \sqrt{x - 5}\). Our goal is to determine which of these functions can produce an output of \(y = -4\). Recall that the square root function \(f(u) = \sqrt{u}\) is defined for \(u \ge 0\) and always returns non-negative values, i.e., \(\sqrt{u} \ge 0\).
For the first function, \(y = \sqrt{x} - 5\), the domain requires \(x \ge 0\). Since \(\sqrt{x} \ge 0\), the minimum value of \(y\) is \(0 - 5 = -5\). Therefore, the range is \([-5, \infty)\). To check if \(-4\) is in this range, we solve \(\sqrt{x} - 5 = -4\):
\[\sqrt{x} = 1\]
\[x = 1\]
Since \(x=1\) is in the domain (\(1 \ge 0\)), this function indeed includes \(-4\) in its range.
For the second function, \(y = \sqrt{x} + 5\), since \(\sqrt{x} \ge 0\), the range is \([5, \infty)\). Clearly, \(-4 < 5\), so \(-4\) is not in the range.
For the third function, \(y = \sqrt{x + 5}\), the range is \([0, \infty)\) because the output of the square root function is always non-negative. Since \(-4 < 0\), this function cannot output \(-4\).
For the fourth function, \(y = \sqrt{x - 5}\), the range is similarly \([0, \infty)\). Since \(-4 < 0\), this function cannot output \(-4\).
Comparing the results, only the function \(y = \sqrt{x} - 5\) has a range that includes the value \(-4\). Thus, the correct function is \(\boxed{y = \sqrt{x} - 5}\).
Understanding the range of a function is crucial in fields like physics or economics, where variables often have physical constraints. For instance, if you are modeling the depth of a submarine (where negative values represent depth below sea level) using a function of time \(t\), knowing the range helps you determine the deepest point the submarine can reach. Just as we checked if -4 is a possible output here, engineers check if their models can accurately represent the physical limitations of their systems.