If $f(x)=\sqrt{x-3}$, which inequality can be used to find the domain of $f(x)$?
The domain of a function consists of all possible real numbers \(x\) for which the function is defined and outputs a real value. For a square root function of the form \(f(x) = \sqrt{u}\), the expression under the square root, known as the radicand, must be non-negative because the square root of a negative number is not a real number. Therefore, the condition for the domain is \(u \geq 0\).
For the given function \(f(x) = \sqrt{x - 3}\), the radicand is \(u = x - 3\). Based on the rule established in the previous step, for the function to produce real outputs, we must have:
\[x - 3 \geq 0\]
This inequality ensures that the value inside the square root is zero or positive, which is exactly what is required for the square root to be well-defined within the set of real numbers.
We compare our derived inequality \(x - 3 \geq 0\) with the provided options:
The derived inequality matches option (b). Therefore, the correct inequality to find the domain of \(f(x) = \sqrt{x-3}\) is \(x - 3 \geq 0\). Thus, the final answer is b.
Imagine you are designing a computer program that calculates the side length of a square based on its area. The area must be zero or positive; you cannot have a square with a negative area. Similarly, when working with mathematical functions, the domain represents the "allowable input values" that keep the result meaningful and within the set of real numbers. Learning to identify these domains helps in real-world applications like data analysis, engineering simulations, or programming, where you must ensure your input values are valid for the formulas you are using.