Question

The domain of \(f(x)=\frac{log_{x+1}(x-2)}{e^{2logx}-(2x+3)}\), \(x∈R\) is

• A. R-{-1,3}
• B. (-1,∞)-{3}
• C. R-{3}
• D. (2,∞)-{3}

Answer 1

The Correct Option is D

Step 1: Analyzing the function for valid values of \( x \). 

We need to check the conditions for the logarithmic and exponential parts:

• The argument of the logarithm \( x - 2 \) must be greater than 0: 
  \( x - 2 > 0 \Rightarrow x > 2 \)
• The argument of the logarithm \( x + 1 \) must be greater than 0: 
  \( x + 1 > 0 \Rightarrow x > -1 \)
• For the denominator \( \frac{1}{x+1} \) to be valid, it must not be 0, so: 
  \( x + 1 \neq 0 \Rightarrow x \neq -1 \) and \( x > 0 \)
• The denominator \( x^2 - 2x - 3 \) must not be 0, so: 
  \( (x - 3)(x + 1) \neq 0 \Rightarrow x \neq -1, 3 \)

Step 2: Combining the conditions.

From the above conditions, we get the domain of the function:

\( x > 2 \) and \( x \neq 3 \)

Thus, the domain of the function is \( (2, \infty) - \{3\} \) .

Step 3: Conclusion.

The correct answer is option (4) \( (2, \infty) - \{3\} \).