Given the function [tex]\( f(t)=\sqrt{3 t-9} \)[/tex], we want to determine the domain of this function.
1. Understanding the Domain:
The function involves a square root, [tex]\(\sqrt{3t - 9}\)[/tex]. The expression inside the square root, [tex]\(3t - 9\)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers.
2. Setting up the Inequality:
To find when the expression inside the square root is non-negative, we set up the following inequality:
[tex]\[
3t - 9 \geq 0
\][/tex]
3. Solving the Inequality:
We solve the inequality step-by-step to find the values of [tex]\( t \)[/tex] for which the inequality holds true.
[tex]\[
3t - 9 \geq 0
\][/tex]
Add 9 to both sides:
[tex]\[
3t \geq 9
\][/tex]
Divide both sides by 3:
[tex]\[
t \geq 3
\][/tex]
4. Conclusion:
The domain of the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex] is all [tex]\( t \)[/tex] such that [tex]\( t \geq 3 \)[/tex].
Thus, the domain of the function [tex]\( f(t) \)[/tex] is [tex]\( t \geq 3 \)[/tex]. The constraint for the domain is [tex]\( t \geq 3 \)[/tex]. This ensures the expression under the square root is non-negative.
So, the final answer is that the domain constraint for the function [tex]\( f(t) \)[/tex] is at least 3.
