Domain of Function:
The given function is F(x) =(e^cos-¹(logx^2))^½. The base of the logarithmic function is 4. We need to find the domain of the function.

Step 1: Define the components of the function:
Here, we can see that the function has two components:
1. e^cos-¹(logx^2)
2. √(e^cos-¹(logx^2))

Step 2: Find the domain of the first component:
The first component of the function is e^cos-¹(logx^2). The range of cos-¹ is [-1,1]. Therefore, the range of e^cos-¹(logx^2) is [e^-1,e^1]. The logarithmic function has a domain of (0,∞). Therefore, the domain of logx^2 is (0,∞). Now, we need to find the domain of cos-¹(logx^2). The range of logx^2 is (−∞,∞). Therefore, the domain of cos-¹(logx^2) is [−1,1]. Hence, the domain of the first component is [e^-1,e^1].

Step 3: Find the domain of the second component:
The second component of the function is √(e^cos-¹(logx^2)). The square root function has a domain of [0,∞). Therefore, we need to find the range of e^cos-¹(logx^2) to determine the domain of the square root function. We know that the range of cos-¹ is [-1,1]. Therefore, the range of e^cos-¹(logx^2) is [e^-1,e^1]. Since the square root function has a domain of [0,∞), the domain of the second component is [e^-1,e^1].

Step 4: Find the domain of the function:
The domain of the function is the intersection of the domains of the two components. Therefore, the domain of the function is [e^-1,e^1].

Summary:
The domain of the given function F(x) =(e^cos-¹(logx^2))^½, where log is at the base of 4, is [e^-1,e^1].