Let `f :[2,oo)to {1,oo)` defined by `f (x)=2^(x ^(4)-4x ^(3))and g : [(pi)/(2), pi] to A ` defined by `g (x) = (sin x+4)/(sin x-2)` be two invertible functions, then
The set "A" equals to

Options

A. `[5,2]`
B. `[-2,5]`
C. `[-5,2]`
D. `[-5,-2]`

Answer

To solve the problem, we need to find the set \( A \) for the function \( g(x) = \frac{\sin x + 4}{\sin x - 2} \) defined on the interval \( \left[\frac{\pi}{2}, \pi\right] \).

### Step-by-Step Solution:

1. **Identify the Function and Interval**:
We have the function \( g(x) = \frac{\sin x + 4}{\sin x - 2} \) defined on the interval \( \left[\frac{\pi}{2}, \pi\right] \).

2. **Evaluate the Function at the Endpoints**:
- Calculate \( g\left(\frac{\pi}{2}\right) \):
\[ g\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right) + 4}{\sin\left(\frac{\pi}{2}\right) - 2} = \frac{1 + 4}{1 - 2} = \frac{5}{-1} = -5 \]
- Calculate \( g(\pi) \):
\[ g(\pi) = \frac{\sin(\pi) + 4}{\sin(\pi) - 2} = \frac{0 + 4}{0 - 2} = \frac{4}{-2} = -2 \]

3. **Determine the Behavior of \( g(x) \)**:
- We need to check if \( g(x) \) is increasing or decreasing in the interval \( \left[\frac{\pi}{2}, \pi\right] \). We can do this by finding the derivative \( g'(x) \).

4. **Find the Derivative \( g'(x) \)**:
Using the quotient rule:
\[ g'(x) = \frac{(\cos x)(\sin x - 2) - (\sin x + 4)(\cos x)}{(\sin x - 2)^2} \]
Simplifying the numerator:
\[ g'(x) = \frac{\cos x (\sin x - 2 - \sin x - 4)}{(\sin x - 2)^2} = \frac{\cos x (-6)}{(\sin x - 2)^2} = \frac{-6 \cos x}{(\sin x - 2)^2} \]

5. **Analyze the Sign of \( g'(x) \)**:
- On the interval \( \left[\frac{\pi}{2}, \pi\right] \), \( \cos x \) is non-positive (it is zero at \( \frac{\pi}{2} \) and negative for \( x \in \left(\frac{\pi}{2}, \pi\right) \)).
- Thus, \( g'(x) \leq 0 \), indicating that \( g(x) \) is decreasing on this interval.

6. **Determine the Range of \( g(x) \)**:
Since \( g(x) \) is decreasing from \( g\left(\frac{\pi}{2}\right) = -5 \) to \( g(\pi) = -2 \), the range of \( g(x) \) is:
\[ g(x) \in (-5, -2) \]

7. **Conclusion**:
Therefore, the set \( A \) is:
\[ A = (-5, -2) \]