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 =

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 \( g: A \to A \) given the conditions \( f(1) = 2 \), \( f(2) = 3 \), \( f(3) = 4 \), \( f(4) = 1 \), \( g(1) = 3 \), and the condition \( f \circ g = g \circ f \).

### Step-by-Step Solution:

1. **Write down the function \( f \)**:
\[ f(1) = 2, \quad f(2) = 3, \quad f(3) = 4, \quad f(4) = 1 \]

2. **Start with the condition \( f \circ g = g \circ f \)**:
This means that for all \( x \in A \), \( f(g(x)) = g(f(x)) \).

3. **Evaluate for \( x = 1 \)**:
\[ f(g(1)) = g(f(1)) \]
We know \( g(1) = 3 \) and \( f(1) = 2 \), so:
\[ f(3) = g(2) \]
From the function \( f \), we know \( f(3) = 4 \). Thus:
\[ g(2) = 4 \]

4. **Evaluate for \( x = 2 \)**:
\[ f(g(2)) = g(f(2)) \]
We have \( g(2) = 4 \) and \( f(2) = 3 \), so:
\[ f(4) = g(3) \]
From the function \( f \), we know \( f(4) = 1 \). Thus:
\[ g(3) = 1 \]

5. **Evaluate for \( x = 3 \)**:
\[ f(g(3)) = g(f(3)) \]
We have \( g(3) = 1 \) and \( f(3) = 4 \), so:
\[ f(1) = g(4) \]
From the function \( f \), we know \( f(1) = 2 \). Thus:
\[ g(4) = 2 \]

6. **Evaluate for \( x = 4 \)**:
\[ f(g(4)) = g(f(4)) \]
We have \( g(4) = 2 \) and \( f(4) = 1 \), so:
\[ f(2) = g(1) \]
From the function \( f \), we know \( f(2) = 3 \). Thus:
\[ g(1) = 3 \]

7. **Summarize the results**:
We have found:
\[ g(1) = 3, \quad g(2) = 4, \quad g(3) = 1, \quad g(4) = 2 \]

8. **Write the function \( g \)**:
The function \( g \) can be summarized as:
\[ g = \{(1, 3), (2, 4), (3, 1), (4, 2)\} \]

### Final Result:
Thus, the function \( g \) is:
\[ g(1) = 3, \quad g(2) = 4, \quad g(3) = 1, \quad g(4) = 2 \]