Let f(x) = px + q and g(x) = mx +n. Then f (g(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(n) =g(p)

Answer

To solve the problem, we need to find the condition under which the compositions of the functions \( f \) and \( g \) are equal, i.e., \( f(g(x)) = g(f(x)) \).

1. **Define the Functions**:
\[ f(x) = px + q \]
\[ g(x) = mx + n \]

2. **Calculate \( f(g(x)) \)**:
Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(mx + n) = p(mx + n) + q = pmx + pn + q \]

3. **Calculate \( g(f(x)) \)**:
Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(px + q) = m(px + q) + n = mpx + mq + n \]

4. **Set the Two Expressions Equal**:
Now, we set \( f(g(x)) \) equal to \( g(f(x)) \):
\[ pmx + pn + q = mpx + mq + n \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ pmx - mpx + pn + q - mq - n = 0 \]
This simplifies to:
\[ (pm - mp)x + (pn + q - mq - n) = 0 \]

6. **Analyze the Coefficients**:
For the equation to hold for all \( x \), both coefficients must equal zero:
- From the coefficient of \( x \):
\[ pm - mp = 0 \quad \Rightarrow \quad pm = mp \]
- From the constant terms:
\[ pn + q - mq - n = 0 \quad \Rightarrow \quad pn + q = mq + n \]

7. **Final Equations**:
Thus, we have two equations:
1. \( pm = mp \) (which is always true for real numbers)
2. \( pn + q = mq + n \)

This means that the condition for \( f(g(x)) = g(f(x)) \) is equivalent to the equation \( pn + q = mq + n \).