Question

if \\(f(x) = \sqrt{\frac{1}{2}x - 10} + 3\\), which inequality can be used to find the domain of \\(f(x)\\)?

• \\(\sqrt{\frac{1}{2}x} \ge 0\\)
• \\(\frac{1}{2}x \ge 0\\)
• \\(\frac{1}{2}x - 10 \ge 0\\)
• \\(\sqrt{\frac{1}{2}x - 10} + 3 \ge 0\\)

Explanation:

Identify the condition for the domain of a square root function

For a real-valued square root function \(g(x) = \sqrt{u(x)}\), the radicand must be non-negative:

$$ u(x) \ge 0 $$

Apply the condition to the given function

Given \(f(x) = \sqrt{\frac{1}{2}x - 10} + 3\), the radicand is:

$$ \frac{1}{2}x - 10 $$

Setting this expression to be non-negative yields:

$$ \frac{1}{2}x - 10 \ge 0 $$

Answer:

• (A) \(\sqrt{\frac{1}{2}x} \ge 0\)
• (B) \(\frac{1}{2}x \ge 0\)
• (C) \(\frac{1}{2}x - 10 \ge 0\) (Correct answer)
• (D) \(\sqrt{\frac{1}{2}x - 10} + 3 \ge 0\)