Questions: Determine the domain of the function. f(x) = sqrt(3x - 1) A. x x ≠ 1/3 B. x x ≥ 1/3 C. x x ≤ 1/3 D. x x is any real number

Determine the domain of the function. f(x) = sqrt(3x - 1) A. x x ≠ 1/3 B. x x ≥ 1/3 C. x x ≤ 1/3 D. x x is any real number
Transcript text: Determine the domain of the function. \[ f(x)=\sqrt{3 x-1} \] A. $\left\{x \left\lvert\, x \neq \frac{1}{3}\right.\right\}$ B. $\left\{x \left\lvert\, x \geq \frac{1}{3}\right.\right\}$ C. $\left\{x \left\lvert\, x \leq \frac{1}{3}\right.\right\}$ D. $\{\mathrm{x} \mid \mathrm{x}$ is any real number $\}$
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Solution

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Solution Steps

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{3x - 1} \). The expression inside the square root, \( 3x - 1 \), must be non-negative for the function to be defined in the real numbers. This is because the square root of a negative number is not a real number.

Step 2: Set Up the Inequality

To find the domain, we need to solve the inequality:

\[ 3x - 1 \geq 0 \]

Step 3: Solve the Inequality

Add 1 to both sides:

\[ 3x \geq 1 \]

Divide both sides by 3:

\[ x \geq \frac{1}{3} \]

Final Answer

The domain of the function is all real numbers \( x \) such that \( x \geq \frac{1}{3} \). Therefore, the correct answer is:

\[ \boxed{\text{B. } \left\{x \left\lvert\, x \geq \frac{1}{3}\right.\right\}} \]

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