The function `f: C -> C` defined by `f(x) = (ax+b)/(cx+d)` for `x in C` where `bd != 0` reduces to a constant function if

Options

A. `a=c`
B. `b=c`
C. `ad=bc`
D. `ab=cd`

Answer

To determine when the function \( f(x) = \frac{ax + b}{cx + d} \) reduces to a constant function, we can follow these steps:

### Step-by-Step Solution:

1. **Set the function equal to a constant**:
\[ f(x) = k \quad \text{(where \( k \) is a constant)} \]
This means:
\[ \frac{ax + b}{cx + d} = k \]

2. **Cross-multiply to eliminate the fraction**:
\[ ax + b = k(cx + d) \]
Expanding the right-hand side gives:
\[ ax + b = kcx + kd \]

3. **Rearranging the equation**:
\[ ax - kcx = kd - b \]
This can be rewritten as:
\[ (a - kc)x = kd - b \]

4. **For \( f(x) \) to be a constant function, the coefficient of \( x \) must be zero**:
This implies:
\[ a - kc = 0 \quad \text{and} \quad kd - b = 0 \]

5. **From \( a - kc = 0 \)**, we can express \( k \):
\[ kc = a \quad \Rightarrow \quad k = \frac{a}{c} \quad \text{(assuming \( c \neq 0 \))} \]

6. **Substituting \( k \) into the second equation \( kd - b = 0 \)**:
\[ \frac{a}{c}d - b = 0 \quad \Rightarrow \quad ad = bc \]

7. **Conclusion**:
The function \( f(x) \) reduces to a constant function if:
\[ ad = bc \]

### Final Answer:
The function \( f(x) = \frac{ax + b}{cx + d} \) reduces to a constant function if \( ad = bc \).