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. \([4,\infty)\)
• C. \((1,\infty)\)
• D. \((1,3)\)

Hint:

For logarithms with base \(0\lt a\lt 1\), the inequality reverses in nature. Also, when a square root is in the denominator, its inside expression must be strictly positive.

Answer 1

The Correct Option is A

Step 1: State both constraints simultaneously.
For \(f(x)=\dfrac{\sqrt{\log_{0.5}(x-3)}}{\sqrt{x-1}}\) to be real: (i) denominator \(\sqrt{x-1}>0\) requires \(x>1\); (ii) inside numerator: \(\log_{0.5}(x-3)\geq 0\) and \(x-3>0\).

Step 2: Solve the log inequality.
Since base \(0.5<1\), \(\log_{0.5}(x-3)\geq 0\) means \(x-3\leq 0.5^0=1\), i.e., \(x\leq 4\). Also \(x-3>0\) gives \(x>3\). Combining all: \(x\in(3,4]\). \[ \boxed{(3,4]} \]