If `f={(1,4),(2,5),(3,6)}` and `g={(4,8),(5,7),(6,9)}`, then gof is

Options

A. `{phi}`
B. `{(1,8),(2,7),(3,9)}`
C. `{(1,7),(2,8),(3,9)}`
D. none of these

Answer

To find \( g \circ f \) (read as "g of f"), we need to evaluate the composition of the two functions \( f \) and \( g \). The function \( f \) maps inputs to outputs, and then we will take those outputs and use them as inputs for the function \( g \).

### Step-by-step Solution:

1. **Identify the Functions**:
- The function \( f \) is given by the set of ordered pairs:
\[ f = \{(1, 4), (2, 5), (3, 6)\} \]
- The function \( g \) is given by the set of ordered pairs:
\[ g = \{(4, 8), (5, 7), (6, 9)\} \]

2. **Evaluate \( g \circ f \)**:
- We will evaluate \( g(f(x)) \) for each \( x \) in the domain of \( f \).

3. **For \( x = 1 \)**:
- From \( f \), \( f(1) = 4 \).
- Now, substitute \( 4 \) into \( g \): \( g(4) = 8 \).
- Thus, \( g(f(1)) = 8 \).
- This gives us the ordered pair \( (1, 8) \).

4. **For \( x = 2 \)**:
- From \( f \), \( f(2) = 5 \).
- Now, substitute \( 5 \) into \( g \): \( g(5) = 7 \).
- Thus, \( g(f(2)) = 7 \).
- This gives us the ordered pair \( (2, 7) \).

5. **For \( x = 3 \)**:
- From \( f \), \( f(3) = 6 \).
- Now, substitute \( 6 \) into \( g \): \( g(6) = 9 \).
- Thus, \( g(f(3)) = 9 \).
- This gives us the ordered pair \( (3, 9) \).

6. **Combine the Results**:
- The final result of \( g \circ f \) is the set of ordered pairs:
\[ g \circ f = \{(1, 8), (2, 7), (3, 9)\} \]

### Final Answer:
\[ g \circ f = \{(1, 8), (2, 7), (3, 9)\} \]