Let's consider the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex].
For the expression under the square root, [tex]\( 3t - 9 \)[/tex], we must ensure that it is non-negative because the square root of a negative number is not defined in the set of real numbers. This means that:
[tex]\[ 3t - 9 \geq 0 \][/tex]
We need to solve the inequality [tex]\( 3t - 9 \geq 0 \)[/tex].
1. First, add 9 to both sides of the inequality to isolate the term with [tex]\( t \)[/tex]:
[tex]\[ 3t - 9 + 9 \geq 0 + 9 \][/tex]
[tex]\[ 3t \geq 9 \][/tex]
2. Next, divide both sides of the inequality by 3 to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{3t}{3} \geq \frac{9}{3} \][/tex]
[tex]\[ t \geq 3 \][/tex]
Thus, the inequality [tex]\( 3t - 9 \)[/tex] must be greater than or equal to zero for the function [tex]\( f(t) \)[/tex] to be defined. Therefore, the correct option is:
[tex]\( 3t - 9 \)[/tex] must be greater than or equal to zero.
