Relations And Functions

Step 1: Understanding the Concept
For the function to be defined, we need to satisfy three conditions based on its structure: 1. The denominator cannot be zero.
2. The expression inside the square root must be non-negative.
3. The argument of the logarithm must be positive.
Combining the first two conditions, the expression inside the square root, , must be strictly positive.
Step 2: Key Formula or Approach
We need to solve the inequality .
Recall the property of logarithms: if and the base is between 0 and 1 ( ), then the inequality sign reverses, i.e., . Also, the argument must be positive, .
Step 3: Detailed Explanation
The main condition for the function to be defined is:
Since the base of the logarithm is (which is between 0 and 1), the inequality sign flips when we remove the logarithm.
Additionally, the argument of any logarithm must be positive:
So, we need to solve the combined inequality: .
We solve this in two parts.
Part 1:
The critical points are and . We can use a sign chart or test intervals.
The numerator is positive for and negative for .
The denominator is positive for and negative for .
The fraction is positive when both have the same sign. This does not happen. It is positive when (num and den ) or (num and den ).
Case (i): and and . So, .
Case (ii): and and . This is impossible.
So, the solution to Part 1 is .
Part 2:
The critical points are and . The fraction is negative when the numerator and denominator have opposite signs.
Case (i): and and . The intersection is .
Case (ii): and and . The intersection is .
So, the solution to Part 2 is or .
Now, we must find the intersection of the solutions from Part 1 and Part 2.
We need such that AND .
The common interval is .
Step 4: Final Answer
The domain of the function is .
The question states the domain is . So, and .
We need to find the value of .
Now we express this result in terms of :
(A)
(B)
(C)
(D)
The correct option is (D).