Let `A = {1,2,3,4} and f : A to ` A satisfy `f (1) =2, f(2)=3, f(3)=4, f (4)=1.` Suppose `g:A to A` satisfies `g (1) =3 and fog = gof , ` then g =
`(a).{(1,3), (2,1), (3,2), (4,4)}` `(b).{(1,3), (2,4),(3,1),(4,2)}` `(c).{(1,3),(2,2),(3,4),(4,3)}` `(d).{(1,3),(2,4),(3,2),(4,1)}`

Options

A. `{(1,3), (2,1), (3,2), (4,4)}`
B. `{(1,3), (2,4),(3,1),(4,2)}`
C. `{(1,3),(2,2),(3,4),(4,3)}`
D. `{(1,3),(2,4),(3,2),(4,1)}`

Answer

To solve the problem, we need to find the function \( g: A \to A \) such that \( g(1) = 3 \) and \( f \circ g = g \circ f \). We will analyze the given function \( f \) and the condition step by step.

### Step 1: Understand the function \( f \)
The function \( f \) is defined as follows:
- \( f(1) = 2 \)
- \( f(2) = 3 \)
- \( f(3) = 4 \)
- \( f(4) = 1 \)

This means that \( f \) is a cyclic permutation of the set \( A \).

### Step 2: Use the condition \( f \circ g = g \circ f \)
The condition \( f \circ g = g \circ f \) means that for any \( x \in A \):
\[ f(g(x)) = g(f(x)) \]

### Step 3: Substitute \( x = 1 \)
Using \( x = 1 \):
\[ f(g(1)) = g(f(1)) \]
Since \( g(1) = 3 \) (given), we have:
\[ f(3) = g(2) \]
From the definition of \( f \):
\[ f(3) = 4 \]
So,
\[ g(2) = 4 \]

### Step 4: Substitute \( x = 2 \)
Using \( x = 2 \):
\[ f(g(2)) = g(f(2)) \]
We already found \( g(2) = 4 \), so:
\[ f(4) = g(3) \]
From the definition of \( f \):
\[ f(4) = 1 \]
Thus,
\[ g(3) = 1 \]

### Step 5: Substitute \( x = 3 \)
Using \( x = 3 \):
\[ f(g(3)) = g(f(3)) \]
We found \( g(3) = 1 \), so:
\[ f(1) = g(4) \]
From the definition of \( f \):
\[ f(1) = 2 \]
Thus,
\[ g(4) = 2 \]

### Step 6: Substitute \( x = 4 \)
Using \( x = 4 \):
\[ f(g(4)) = g(f(4)) \]
We found \( g(4) = 2 \), so:
\[ f(2) = g(1) \]
From the definition of \( f \):
\[ f(2) = 3 \]
Thus,
\[ g(1) = 3 \] (which is consistent with the given condition).

### Step 7: Compile the results
Now we have:
- \( g(1) = 3 \)
- \( g(2) = 4 \)
- \( g(3) = 1 \)
- \( g(4) = 2 \)

This gives us the mapping:
\[ g = \{(1, 3), (2, 4), (3, 1), (4, 2)\} \]

### Conclusion
The function \( g \) is:
\[ g = \{(1, 3), (2, 4), (3, 1), (4, 2)\} \]

Thus, the correct answer is:
**(b) \{(1, 3), (2, 4), (3, 1), (4, 2)\}**