What is the domain of this square root function?
$\begin{array}{c}
f(x)=\sqrt{7 x-42} \\
x \geq[?]
\end{array}$
The function given is \(f(x) = \sqrt{7x - 42}\). For a square root function of the form \(\sqrt{u(x)}\) to be defined within the set of real numbers, the expression under the square root must be non-negative. This is because the square root of a negative number is not a real number. Therefore, we must satisfy the inequality \(7x - 42 \geq 0\).
To find the domain, we need to solve the inequality \(7x - 42 \geq 0\). Adding 42 to both sides of the inequality gives \(7x \geq 42\). Next, we divide both sides by 7 to isolate \(x\): \(x \geq \frac{42}{7}\), which simplifies to \(x \geq 6\).
The inequality \(x \geq 6\) defines the domain of the function. Any value of \(x\) that is less than 6 would make the expression \(7x - 42\) negative, resulting in an undefined value for the square root in the set of real numbers. Therefore, the domain of \(f(x) = \sqrt{7x - 42}\) is \(x \geq 6\). The missing value in the bracket is 6.
Understanding the domain of square root functions is essential in various fields, such as engineering and physics, where square roots often appear in formulas for speed, distance, or time. For example, if you are calculating the time \(t\) it takes for an object to reach a certain distance under constant acceleration, the formula might include a square root. Knowing the domain ensures that your calculations remain valid and physically meaningful, preventing errors that arise from attempting to process impossible scenarios, like taking the square root of a negative time or distance.