To find the largest interval in which the function \( f(x) = 4^{-x^2} + \cos^{-1}(x/2 - 1) + \log(\cos x) \) is defined, we need to analyze the domain of each component of the function.
### Step 1: Analyze the first component \( 4^{-x^2} \)
The function \( 4^{-x^2} \) is defined for all real numbers \( x \). Therefore, there are no restrictions from this component.
**Hint:** The exponential function is defined for all real numbers.
### Step 2: Analyze the second component \( \cos^{-1}(x/2 - 1) \)
The function \( \cos^{-1}(y) \) is defined for \( y \) in the interval \([-1, 1]\). Thus, we need to solve the inequality:
\[
-1 \leq \frac{x}{2} - 1 \leq 1
\]
**Hint:** Remember that the range of the inverse cosine function is limited to \([-1, 1]\).
### Step 3: Solve the inequalities for \( \cos^{-1}(x/2 - 1) \)
1. **Lower Bound:**
\[
\frac{x}{2} - 1 \geq -1 \implies \frac{x}{2} \geq 0 \implies x \geq 0
\]
2. **Upper Bound:**
\[
\frac{x}{2} - 1 \leq 1 \implies \frac{x}{2} \leq 2 \implies x \leq 4
\]
Combining these results, we have:
\[
0 \leq x \leq 4
\]
**Hint:** When solving inequalities, ensure to combine the results correctly to find the valid range.
### Step 4: Analyze the third component \( \log(\cos x) \)
The logarithmic function \( \log(z) \) is defined for \( z > 0 \). Therefore, we need \( \cos x > 0 \).
The cosine function is positive in the intervals:
- \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) (first quadrant)
- \( (0, \frac{\pi}{2}) \) (fourth quadrant)
Thus, within the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), we need to find where \( \cos x > 0 \).
**Hint:** Identify the intervals where the cosine function is positive based on its periodic nature.
### Step 5: Combine the intervals
From the analysis:
- From \( \cos^{-1}(x/2 - 1) \), we found \( 0 \leq x \leq 4 \).
- From \( \log(\cos x) \), we found that \( x \) must be in \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).
The intersection of these two intervals is:
\[
[0, 4] \cap \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)
\]
Since \( \frac{\pi}{2} \approx 1.57 \), the combined interval is:
\[
[0, \frac{\pi}{2})
\]
### Final Answer
Thus, the largest interval for which the function \( f(x) \) is defined is:
\[
\boxed{[0, \frac{\pi}{2})}
\]
