To determine the domain of the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex], we need to make sure that the expression inside the square root, [tex]\( 3t - 9 \)[/tex], is non-negative (i.e., greater than or equal to zero). This is because the square root function is only defined for non-negative numbers.
Here are the steps to find the domain:
1. Define the inequality:
[tex]\[
3t - 9 \geq 0
\][/tex]
2. Solve the inequality for [tex]\( t \)[/tex]:
- Add 9 to both sides to isolate the term with [tex]\( t \)[/tex]:
[tex]\[
3t \geq 9
\][/tex]
- Divide both sides by 3 to solve for [tex]\( t \)[/tex]:
[tex]\[
t \geq \frac{9}{3}
\][/tex]
[tex]\[
t \geq 3
\][/tex]
So, for the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex] to be defined, [tex]\( t \)[/tex] must be greater than or equal to 3.
Therefore, the domain of the function is:
[tex]\[
t \geq 3
\][/tex]
In interval notation, the domain can be expressed as:
[tex]\[
[3, \infty)
\][/tex]
Thus, the minimum value of [tex]\( t \)[/tex] for which the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex] is defined is [tex]\( 3 \)[/tex].
