If `f(x)=ax+bandg(x)=cx+d` such that `f[g(x)]=g[f(x)]` then which one of the following is correct?

Options

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

Explanation

To solve the problem, we need to analyze the functions \( f(x) = ax + b \) and \( g(x) = cx + d \) given that \( f[g(x)] = g[f(x)] \). We will find the expressions for \( f[g(x)] \) and \( g[f(x)] \) and set them equal to each other.

### Step-by-Step Solution:

1. **Define the Functions:**
\[ f(x) = ax + b \]
\[ g(x) = cx + d \]

2. **Calculate \( f[g(x)] \):**
We substitute \( g(x) \) into \( f(x) \):
\[ f[g(x)] = f(cx + d) = a(cx + d) + b \]
Expanding this gives:
\[ f[g(x)] = acx + ad + b \]

3. **Calculate \( g[f(x)] \):**
We substitute \( f(x) \) into \( g(x) \):
\[ g[f(x)] = g(ax + b) = c(ax + b) + d \]
Expanding this gives:
\[ g[f(x)] = acx + bc + d \]

4. **Set the Two Expressions Equal:**
Since we know \( f[g(x)] = g[f(x)] \), we set the two expressions equal:
\[ acx + ad + b = acx + bc + d \]

5. **Cancel the \( acx \) Terms:**
By canceling \( acx \) from both sides, we have:
\[ ad + b = bc + d \]

6. **Rearranging the Equation:**
Rearranging gives us:
\[ ad - d = bc - b \]
Factoring both sides results in:
\[ d(a - 1) = b(c - 1) \]

7. **Analyzing the Options:**
We need to check which of the given options is correct based on the derived relationship.

- Option 1: \( f(c) = g(a) \)
- Option 2: \( f(a) = g(c) \)
- Option 3: \( f(c) = g(d) \)
- Option 4: \( f(d) = g(b) \)

We will evaluate \( f(d) \) and \( g(b) \):
\[ f(d) = a(d) + b = ad + b \]
\[ g(b) = c(b) + d = bc + d \]
From our earlier equation, we know:
\[ ad + b = bc + d \]
Thus, we conclude:
\[ f(d) = g(b) \]

### Final Answer:
The correct option is:
\[ \text{Option 4: } f(d) = g(b) \]