If f (x) = ax +b and g(x) = x+d then (fog) x=(gof) x implies

Options

A. `f(a) = g( c)`
B. ` f(d) = g(b)`
C. `f(b)=g(a)`
D. `f(c )= g(a)`

Answer

To solve the problem, we need to establish the equality of the two composite functions \( f(g(x)) \) and \( g(f(x)) \) given the functions \( f(x) = ax + b \) and \( g(x) = x + d \).

### Step-by-Step Solution:

1. **Identify the Functions**:
- We have \( f(x) = ax + b \)
- We have \( g(x) = x + d \)

2. **Calculate \( f(g(x)) \)**:
- Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x + d) = a(x + d) + b \]
- Simplifying this gives:
\[ f(g(x)) = ax + ad + b \]

3. **Calculate \( g(f(x)) \)**:
- Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(ax + b) = (ax + b) + d \]
- Simplifying this gives:
\[ g(f(x)) = ax + b + d \]

4. **Set the Two Expressions Equal**:
- Now we set \( f(g(x)) \) equal to \( g(f(x)) \):
\[ ax + ad + b = ax + b + d \]

5. **Cancel Common Terms**:
- We can cancel \( ax \) and \( b \) from both sides:
\[ ad = d \]

6. **Solve for \( a \)**:
- Rearranging gives:
\[ ad - d = 0 \implies d(a - 1) = 0 \]
- This implies either \( d = 0 \) or \( a - 1 = 0 \). Therefore:
\[ a = 1 \quad \text{(if \( d \neq 0 \))} \]

### Conclusion:
From the above steps, we conclude that \( a = 1 \) is a necessary condition for the equality \( f(g(x)) = g(f(x)) \) to hold true.