Let `f(x)=px+qandg(x)=mx+n." Then "f(f(x))=g(f(x))` is equivalent to

Options

A. `f(p)=g(m)`
B. `f(q)=g(n)`
C. `f(n)=g(q)`
D. `f(m)=g(p)`

Explanation

To solve the equation \( f(f(x)) = g(f(x)) \) given \( f(x) = px + q \) and \( g(x) = mx + n \), we will follow these steps:

### Step 1: Calculate \( f(f(x)) \)

First, we need to find \( f(f(x)) \):
\[ f(f(x)) = f(px + q) \]
Substituting \( px + q \) into \( f(x) \):
\[ f(px + q) = p(px + q) + q = p^2x + pq + q \]
Thus, we have:
\[ f(f(x)) = p^2x + (pq + q) \]

### Step 2: Calculate \( g(f(x)) \)

Next, we calculate \( g(f(x)) \):
\[ g(f(x)) = g(px + q) \]
Substituting \( px + q \) into \( g(x) \):
\[ g(px + q) = m(px + q) + n = mpx + mq + n \]
Thus, we have:
\[ g(f(x)) = mpx + (mq + n) \]

### Step 3: Set the two expressions equal

Now, we set the two expressions equal to each other:
\[ f(f(x)) = g(f(x)) \]
This gives us:
\[ p^2x + (pq + q) = mpx + (mq + n) \]

### Step 4: Compare coefficients

To solve for the coefficients, we compare the coefficients of \( x \) and the constant terms on both sides:
1. Coefficient of \( x \):
\[ p^2 = mp \]
2. Constant terms:
\[ pq + q = mq + n \]

### Step 5: Rearranging the equations

From the first equation \( p^2 = mp \), we can rearrange it:
\[ p^2 - mp = 0 \quad \Rightarrow \quad p(p - m) = 0 \]
This gives us two cases:
1. \( p = 0 \)
2. \( p = m \)

From the second equation \( pq + q = mq + n \), we can rearrange it:
\[ pq + q - mq - n = 0 \quad \Rightarrow \quad (p - m)q + (q - n) = 0 \]

### Step 6: Analyzing the second equation

For the second equation to hold, we have two cases:
1. If \( p \neq m \), then \( q = n \).
2. If \( p = m \), then this equation holds for any \( q \) and \( n \).

### Conclusion

Thus, the condition \( f(f(x)) = g(f(x)) \) is equivalent to:
1. \( p = 0 \) or \( p = m \)
2. If \( p \neq m \), then \( q = n \).