Answer :
To determine the domain of the function [tex]\(g(x) = \sqrt{3x - 12}\)[/tex], we need to ensure that the expression inside the square root, [tex]\(3x - 12\)[/tex], is non-negative. This is because the square root function is defined only for non-negative values.
### Step-by-Step Solution
1. Identify the expression inside the square root:
[tex]\[ 3x - 12 \][/tex]
2. Set up the inequality for the expression to be non-negative:
[tex]\[ 3x - 12 \geq 0 \][/tex]
3. Solve the inequality for [tex]\(x\)[/tex]:
- Add 12 to both sides of the inequality:
[tex]\[ 3x - 12 + 12 \geq 0 + 12 \][/tex]
[tex]\[ 3x \geq 12 \][/tex]
- Divide both sides by 3:
[tex]\[ \frac{3x}{3} \geq \frac{12}{3} \][/tex]
[tex]\[ x \geq 4 \][/tex]
4. State the domain in interval notation:
- Since [tex]\(x\)[/tex] must be greater than or equal to 4, the domain includes all [tex]\(x\)[/tex] from 4 to infinity.
Therefore, the domain of the function [tex]\(g(x) = \sqrt{3x - 12}\)[/tex] is:
[tex]\[ [4, \infty) \][/tex]
### Step-by-Step Solution
1. Identify the expression inside the square root:
[tex]\[ 3x - 12 \][/tex]
2. Set up the inequality for the expression to be non-negative:
[tex]\[ 3x - 12 \geq 0 \][/tex]
3. Solve the inequality for [tex]\(x\)[/tex]:
- Add 12 to both sides of the inequality:
[tex]\[ 3x - 12 + 12 \geq 0 + 12 \][/tex]
[tex]\[ 3x \geq 12 \][/tex]
- Divide both sides by 3:
[tex]\[ \frac{3x}{3} \geq \frac{12}{3} \][/tex]
[tex]\[ x \geq 4 \][/tex]
4. State the domain in interval notation:
- Since [tex]\(x\)[/tex] must be greater than or equal to 4, the domain includes all [tex]\(x\)[/tex] from 4 to infinity.
Therefore, the domain of the function [tex]\(g(x) = \sqrt{3x - 12}\)[/tex] is:
[tex]\[ [4, \infty) \][/tex]
