- The condition for the square root function \(\sqrt{5x - 5} + 1\) to be defined is that the radicand must be non-negative: \(5x - 5 \geq 0\).
- This corresponds to option a.
- Solving \(5x - 5 \geq 0\) for \(x\) gives \(x \geq 1\).
- The domain is \(x \geq \boxed{1}\).
Explanation
- Analyzing the Domain Condition
To find the domain of the function \(f(x) = \sqrt{5x - 5} + 1\), we must consider the restrictions imposed by the square root. The square root function \(\sqrt{g(x)}\) is defined for real numbers if and only if the expression inside the radical, \(g(x)\), is non-negative. In this case, \(g(x) = 5x - 5\). Therefore, the condition to find the domain is that \(5x - 5 \geq 0\). Comparing this to the given options, we identify that option a, \(5x - 5 \geq 0\), is the correct inequality to use.
- Solving the Inequality
To determine the domain, we solve the inequality \(5x - 5 \geq 0\). Adding 5 to both sides gives \(5x \geq 5\). Dividing both sides by 5, we obtain \(x \geq 1\). Thus, the function is defined for all \(x\) greater than or equal to 1. The domain of the function is \(x \geq 1\).
Examples
Understanding the domain of a function is crucial in real-world applications where certain variables have physically meaningful ranges. For example, if a function represents the time it takes for a ball to reach a certain height, the expression under the square root must be non-negative because time cannot be a negative value inside a square root in this context. Determining the valid input range helps in defining the operational limits of systems, like in engineering, where you must ensure that variables stay within bounds that prevent undefined or physically impossible results.