- Establish the condition that the radicand must be non-negative: \(3t - 9 \ge 0\).
- Solve the inequality by adding 9 and dividing by 3: \(t \ge 3\).
- Express the domain in interval notation: \([3, \infty)\).
- Final Answer: The domain of the function is \(\boxed{[3, \infty)}\).
Explanation
- Establishing the Constraint
To determine the domain of the function \(f(t) = \sqrt{3t - 9}\), we must ensure that the expression inside the square root is non-negative. This is because the square root of a negative number is not a real number. Therefore, we set up the following inequality:
\[3t - 9 \ge 0\]
- Solving the Inequality
We begin with the inequality:
\[3t - 9 \ge 0\]
To isolate the variable \(t\), we first add \(9\) to both sides of the inequality:
\[3t \ge 9\]
Next, we divide both sides by \(3\):
\[t \ge 3\]
This inequality defines all values of \(t\) for which the function is defined.
- Stating the Domain
The inequality \(t \ge 3\) indicates that the function is defined for all real numbers \(t\) that are greater than or equal to \(3\). In interval notation, this domain is represented as \([3, \infty)\). Therefore, the domain of the function \(f(t) = \sqrt{3t - 9}\) is \([3, \infty)\).
Examples
In real-world applications, domains of functions are crucial for understanding valid input ranges. For example, if a function represents the cost of producing a certain number of items, the variable representing the quantity of items cannot be negative. If a function is defined as \(f(q) = \sqrt{5q - 20}\), calculating its domain ensures that we are only considering physically possible quantities of items that lead to real, meaningful cost values, helping in budget planning and inventory management.