For real x,y such that `f(x) ne 0`, if f(x+y) = f(x).f(y), what's the value of f(100x)?

Options

A. `(f(x))^100`
B. `(f(x))^99`
C. `99(f(x))^100`
D. `((f(x))^99)/100`

Answer

To solve the problem, we start with the functional equation given:

1. **Given**: \( f(x+y) = f(x) \cdot f(y) \) for all real \( x, y \) such that \( f(x) \neq 0 \).

We need to find the value of \( f(100x) \).

2. **Step 1**: Let's express \( f(100x) \) in terms of smaller arguments. We can write:
\[ f(100x) = f(99x + x) = f(99x) \cdot f(x) \]

3. **Step 2**: Now, we can express \( f(99x) \) in a similar way:
\[ f(99x) = f(98x + x) = f(98x) \cdot f(x) \]

4. **Step 3**: Substituting \( f(99x) \) back into our expression for \( f(100x) \):
\[ f(100x) = f(98x) \cdot f(x) \cdot f(x) = f(98x) \cdot f(x)^2 \]

5. **Step 4**: Continuing this process, we can express \( f(98x) \) as:
\[ f(98x) = f(97x + x) = f(97x) \cdot f(x) \]
Substituting this into our expression for \( f(100x) \):
\[ f(100x) = f(97x) \cdot f(x) \cdot f(x)^2 = f(97x) \cdot f(x)^3 \]

6. **Step 5**: We can keep doing this until we reach \( f(0) \):
\[ f(100x) = f(0) \cdot f(x)^{100} \]

7. **Step 6**: Since \( f(0) \) can be determined from the functional equation, we can find that \( f(0) = f(0 + 0) = f(0) \cdot f(0) \). This implies \( f(0) = 1 \) (since \( f(x) \neq 0 \)).

8. **Final Step**: Therefore, substituting \( f(0) = 1 \) into our expression gives:
\[ f(100x) = 1 \cdot f(x)^{100} = f(x)^{100} \]

Thus, the value of \( f(100x) \) is \( f(x)^{100} \).