Let f(x) = ax + b and g(x)= cx+ d. Then f (g(x)) = g (fex) is equivalent to:

Options

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

Answer

To solve the problem, we need to find the equivalent expression for the equation \( f(g(x)) = g(f(x)) \) where \( f(x) = ax + b \) and \( g(x) = cx + d \).

### Step-by-Step Solution:

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

2. **Find \( f(g(x)) \)**:
- Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(cx + d) = a(cx + d) + b \]
- Distributing \( a \):
\[ = acx + ad + b \]

3. **Find \( g(f(x)) \)**:
- Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(ax + b) = c(ax + b) + d \]
- Distributing \( c \):
\[ = acx + cb + d \]

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

5. **Cancel \( acx \) from Both Sides**:
- Since \( acx \) appears on both sides, we can cancel it:
\[ ad + b = cb + d \]

6. **Rearrange the Equation**:
- Rearranging gives us:
\[ ad - d = cb - b \]
- Factor out common terms:
\[ d(a - 1) = b(c - 1) \]

### Conclusion:
The equation \( f(g(x)) = g(f(x)) \) is equivalent to:
\[ d(a - 1) = b(c - 1) \]