If f(xy) = f(x)f(y), then f(t) may be of the form

Options

A. t+k
B. ct+k
C. t+c
D. t

Answer

To solve the problem, we need to analyze the functional equation given by \( f(xy) = f(x)f(y) \). This type of equation is characteristic of exponential functions. Let's derive the possible forms of \( f(t) \).

### Step-by-Step Solution:

1. **Understanding the Functional Equation**:
The equation \( f(xy) = f(x)f(y) \) suggests that the function \( f \) has a multiplicative property. This is a common property of exponential functions.

2. **Testing Simple Functions**:
Let's test if \( f(t) = t^k \) (where \( k \) is a constant) could be a solution. We substitute \( f(x) = x^k \) and \( f(y) = y^k \) into the equation:
\[ f(xy) = (xy)^k = x^k y^k = f(x)f(y) \]
This holds true for any real number \( k \).

3. **Considering the Exponential Function**:
Another common form that satisfies this equation is the exponential function. Let’s test \( f(t) = a^t \) (where \( a \) is a positive constant):
\[ f(xy) = a^{xy} = a^x a^y = f(x)f(y) \]
This also holds true.

4. **Conclusion**:
From our analysis, we can conclude that the function \( f(t) \) may take the form of:
- \( f(t) = t^k \) for some constant \( k \)
- \( f(t) = a^t \) for some positive constant \( a \)

### Final Answer:
Thus, the forms that \( f(t) \) may take are:
- \( f(t) = t^k \) (for any real number \( k \))
- \( f(t) = a^t \) (for any positive constant \( a \))