- Identify the condition for the square root: the radicand \(x-1\) must be non-negative.
- Solve the inequality \(x - 1 \geq 0\) to find \(x \geq 1\).
- Express the domain in interval notation as \([1, \infty)\).
- Final answer: \(\boxed{[1, \infty)}\).
Explanation
- Analyzing the Function and Defining the Condition
The given function is \(p(x) = \sqrt{x-1} + 2\). To determine the domain of this function, we must identify all real numbers \(x\) for which the expression \(p(x)\) is well-defined. In the expression \(\sqrt{x-1} + 2\), the number \(2\) is defined for all real numbers. The term \(\sqrt{x-1}\) involves a square root, which is defined in the set of real numbers if and only if the expression inside the root—the radicand—is non-negative. Therefore, we require the radicand \(x-1\) to be greater than or equal to zero.
- Solving the Inequality and Identifying the Domain
To find the domain, we solve the inequality \(x - 1 \geq 0\). Adding \(1\) to both sides of the inequality gives us:
\[x \geq 1\]
This means the function \(p(x)\) is defined for all values of \(x\) that are greater than or equal to \(1\). In interval notation, this set of numbers is written as \([1, \infty)\). Comparing this result to the given options, we see that it corresponds exactly to the second option.
- Final Conclusion
The domain of the function \(p(x) = \sqrt{x-1} + 2\) is \([1, \infty)\), which is the second option listed. Therefore, the correct choice is \(\boxed{[1, \infty)}\).
Examples
Understanding the domain of functions is crucial in real-world applications, such as in engineering or economics, where variables often have restricted ranges. For example, if a function represents the cost of producing \(x\) units of a product, the number of units \(x\) must be at least zero (often defined by a non-negative domain). Similarly, in physical models like calculating the speed of an object moving through a medium where the square root of a force is involved, the domain ensures that the calculations remain within the realm of possible physical realities, avoiding imaginary results that could indicate an impossible state for the system.