To determine the domain of the function [tex]\( f(x) = \sqrt{3x - 27} \)[/tex], we need to ensure that the expression inside the square root is non-negative, since the square root of a negative number is not defined in the real number system.
Follow these steps to find the domain:
1. Set up the inequality:
We need the expression inside the square root, [tex]\( 3x - 27 \)[/tex], to be greater than or equal to zero:
[tex]\[
3x - 27 \geq 0
\][/tex]
2. Solve the inequality:
Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side of the inequality.
[tex]\[
3x - 27 \geq 0
\][/tex]
Add 27 to both sides:
[tex]\[
3x \geq 27
\][/tex]
Divide both sides by 3:
[tex]\[
x \geq 9
\][/tex]
3. Interpret the solution:
The inequality [tex]\( x \geq 9 \)[/tex] means that [tex]\( x \)[/tex] can be any real number starting from 9 and going to positive infinity.
4. Express the domain:
The domain of the function [tex]\( f(x) = \sqrt{3x - 27} \)[/tex] is all [tex]\( x \)[/tex] values that are greater than or equal to 9. In interval notation, this is written as:
[tex]\[
[9, \infty)
\][/tex]
5. Compare with the given options:
The correct option that matches [tex]\([9, \infty)\)[/tex] is (D).
Thus, the domain of the function [tex]\( f(x) = \sqrt{3x - 27} \)[/tex] is:
[tex]\[
\boxed{[9, \infty)}
\][/tex]
