Question

The domain of the real-valued function \(f(x) = \frac{\sqrt{\log_{0.5}(x-3)}}{\sqrt{x-1}}\) is:

• A. \((3, 4]\)
• B. \([4, 6)\)
• C. \((1, 6)\)
• D. \((1, 3)\)

Hint:

Domain Finding}
For logarithmic functions, ensure argument>0
For square roots, ensure expression \(\geq 0\)
Consider all constraints simultaneously

Answer 1

The Correct Option is A

Step 1: Denominator constraint (\(x-1>0\)): \[ x>1 \] Step 2: Numerator constraints: \begin{itemize} \item \(\log_{0.5}(x-3)\) defined when \(x-3>0 \Rightarrow x>3\) \item \(\log_{0.5}(x-3) \geq 0\) (since under square root) \end{itemize} Step 3: Solve \(\log_{0.5}(x-3) \geq 0\): \[ 0<x-3 \leq 1 \Rightarrow 3<x \leq 4 \] Step 4: Combine constraints: \[ 3<x \leq 4 \]