`f: { 1, 2, 3, 4} -> {1, 4, 9, 16} and g: {1, 4. 9, 16) ->{1,1/2,1/3,1/4}` are two bijective functions such that `x_1 gt x_2 => f(x_1) lt f(x_2),g(x_1) gt g(x_2)` then `f^-1(g^-1(1/2))` is equal to

Options

A. 1
B. 4
C. 16
D. 2

Explanation

To solve the problem step by step, we will first analyze the functions \( f \) and \( g \) and then find the required value of \( f^{-1}(g^{-1}(1/2)) \).

### Step 1: Define the functions \( f \) and \( g \)

The function \( f \) is defined as:
- \( f: \{1, 2, 3, 4\} \to \{1, 4, 9, 16\} \)

Since \( f \) is a decreasing function, we can assign values based on the decreasing order:
- \( f(1) = 16 \)
- \( f(2) = 9 \)
- \( f(3) = 4 \)
- \( f(4) = 1 \)

The mapping can be summarized as:
- \( f(1) = 16 \)
- \( f(2) = 9 \)
- \( f(3) = 4 \)
- \( f(4) = 1 \)

The function \( g \) is defined as:
- \( g: \{1, 4, 9, 16\} \to \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\} \)

Since \( g \) is an increasing function, we can assign values based on the increasing order:
- \( g(1) = \frac{1}{4} \)
- \( g(4) = \frac{1}{3} \)
- \( g(9) = \frac{1}{2} \)
- \( g(16) = 1 \)

The mapping can be summarized as:
- \( g(1) = \frac{1}{4} \)
- \( g(4) = \frac{1}{3} \)
- \( g(9) = \frac{1}{2} \)
- \( g(16) = 1 \)

### Step 2: Find \( g^{-1}(1/2) \)

To find \( g^{-1}(1/2) \), we look for the input that gives us \( \frac{1}{2} \):
- From the mapping of \( g \), we see that \( g(9) = \frac{1}{2} \).

Thus, we have:
\[ g^{-1}(1/2) = 9 \]

### Step 3: Find \( f^{-1}(9) \)

Next, we need to find \( f^{-1}(9) \). We look for the input that gives us \( 9 \):
- From the mapping of \( f \), we see that \( f(2) = 9 \).

Thus, we have:
\[ f^{-1}(9) = 2 \]

### Final Result

Combining the results, we find:
\[ f^{-1}(g^{-1}(1/2)) = f^{-1}(9) = 2 \]

### Answer
The final answer is \( 2 \).

---