Find the domain of the function [tex]f(x)=\sqrt{3x-27}[/tex].

A. [tex]\((9, \infty)\)[/tex]
B. [tex]\((-9, \infty)\)[/tex]
C. [tex]\([-9, \infty)\)[/tex]
D. [tex]\([9, \infty)\)[/tex]



Answer :

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]