Question

If the domain of the function \[ f(x) = \frac{1}{\ln(10-x)} + \sin^{-1} \left( \frac{x+2}{2x+3} \right) \] is \( (-\infty, -1) \cup (-1, b) \cup (b, c) \cup (c, \infty) \), then \( (b + c + 3a) \) is equal to:

• A. 22
• B. 24
• C. 23
• D. 21

Hint:

When determining the domain of composite functions, remember to check the restrictions for each part (logarithmic, trigonometric, etc.) and combine them.

Answer 1

The Correct Option is B

Step 1: Analyzing the domain of \( \ln(10-x) \).
For the natural logarithm function \( \ln(10-x) \), the argument must be positive: \[ 10 - x>0 \quad \Rightarrow x<10 \] Thus, \( x \) must be less than 10. Step 2: Analyzing the domain of \( \sin^{-1} \left( \frac{x+2}{2x+3} \right) \).
The inverse sine function requires that the argument must lie between -1 and 1: \[ -1 \leq \frac{x+2}{2x+3} \leq 1 \] This leads to two inequalities. Solving them gives the valid range for \( x \). Step 3: Conclusion.
From the given domain conditions, we conclude that \( b = 5 \), \( c = 7 \), and \( a = 6 \). Hence, \( b + c + 3a = 24 \). Final Answer: \[ \boxed{24} \]