What is the domain of this square root function?
$\begin{array}{c}
f(x)=\sqrt{x+10}+2 \\
x \geq[?]
\end{array}$
The function provided is \(f(x) = \sqrt{x+10} + 2\). To find its domain, we must ensure that the function produces real-valued outputs. A square root function \(\sqrt{u}\) is only defined for real numbers when the radicand \(u\) is greater than or equal to zero. Therefore, we require the expression inside the square root to satisfy the inequality \(x + 10 \geq 0\).
To determine the range of values for \(x\), we solve the linear inequality \(x + 10 \geq 0\). By subtracting 10 from both sides, we isolate \(x\):
\[x + 10 - 10 \geq 0 - 10\]
\[x \geq -10\]
This result indicates that the function is defined for all values of \(x\) greater than or equal to -10.
The domain of the function \(f(x) = \sqrt{x+10} + 2\) is the set of all real numbers \(x\) such that \(x \geq -10\). Thus, the value to fill in the bracketed expression is \(-10\). The final domain is \(\boxed{x \geq -10}\).
Square root functions are used in many real-world fields, such as physics and engineering. For example, if you are modeling the speed of an object moving under certain gravitational forces or calculating the distance of a point from a fixed coordinate, you might encounter expressions like \(d = \sqrt{x^2 + y^2}\). Knowing the domain of these functions ensures that your calculations remain within the physically possible range, preventing errors like taking the square root of a negative number.