To solve the problem step by step, we will break down the components of the question and find the values of \( A \) and \( B \), as well as analyze the functions \( f(x) \) and \( g(x) \).
### Step 1: Calculate \( A \)
Given:
\[
A = \sin^{-1}(\sin 3) + \sin^{-1}(\sin 4) + \sin^{-1}(\sin 5)
\]
Using the property of the inverse sine function:
\[
\sin^{-1}(\sin x) = x \quad \text{if } x \in [-\frac{\pi}{2}, \frac{\pi}{2}]
\]
Otherwise, we adjust \( x \) to find the equivalent angle within the range.
1. **Calculate \( \sin^{-1}(\sin 3) \)**:
- \( 3 \) is in the range \( [0, \pi] \), so:
\[
\sin^{-1}(\sin 3) = 3
\]
2. **Calculate \( \sin^{-1}(\sin 4) \)**:
- \( 4 \) is in the range \( [0, \pi] \), so:
\[
\sin^{-1}(\sin 4) = \pi - 4
\]
3. **Calculate \( \sin^{-1}(\sin 5) \)**:
- \( 5 \) is in the range \( [0, \pi] \), so:
\[
\sin^{-1}(\sin 5) = \pi - 5
\]
Now, substituting these values into \( A \):
\[
A = 3 + (\pi - 4) + (\pi - 5) = 2\pi - 6
\]
### Step 2: Find the value of \( A \)
To simplify:
\[
A = 2\pi - 6 \approx 6.283 - 6 = 0.283 \quad \text{(approximately)}
\]
### Step 3: Analyze the function \( f(x) \)
Given:
\[
f(x) = e^{x^2 + |x|}
\]
**Domain of \( f(x) \)**:
The domain of \( f(x) \) is given as \( [A, \infty) \). Since we have calculated \( A \approx 0.283 \), the domain is:
\[
[A, \infty) = [2\pi - 6, \infty)
\]
**Range of \( f(x) \)**:
Since \( f(x) \) is an exponential function, it takes values from \( e^{A^2 + |A|} \) to \( \infty \).
### Step 4: Find \( B \)
To find \( B \):
1. Calculate \( f(A) \):
\[
f(A) = e^{(2\pi - 6)^2 + |2\pi - 6|}
\]
Since \( 2\pi - 6 \) is negative, we take the absolute value:
\[
|2\pi - 6| = 6 - 2\pi
\]
Thus:
\[
f(A) = e^{(2\pi - 6)^2 + (6 - 2\pi)}
\]
2. Set \( B = f(A) \) and find the minimum value of \( f(x) \) to determine \( B \).
### Step 5: Analyze the function \( g(x) \)
Given:
\[
g(x) = \left(4\cos^4 x - 2\cos 2x - \frac{1}{4}\cos 4x - x^7\right)^{\frac{1}{7}}
\]
**Domain and Range**:
The domain and range of \( g(x) \) is the set of real numbers, which means it can take any real value.
### Step 6: Verify the statements
1. **Check if \( A = -2 \)**:
- We calculated \( A \approx 0.283 \), so this statement is **incorrect**.
2. **Check if \( A + B = -1 \)**:
- We need to calculate \( B \) to verify this.
3. **Check if \( f(g(g(B))) = e^2 \)**:
- This requires evaluating \( g(B) \) and then substituting into \( f \).
4. **Check if both \( f(x) \) and \( g(x) \) are non-invertible**:
- Since \( f(x) \) is not one-to-one, it is non-invertible.
### Conclusion
The correct answers based on our calculations and analysis are:
- \( A \) is not equal to -2.
- \( A + B \) needs verification.
- \( f(g(g(B))) = e^2 \) needs verification.
- Both functions are non-invertible.
