If f is a periodic function and g is a non-periodic function , then

Options

A. fog is always periodic
B. gof is never periodic
C. gof is always periodic
D. none of these

Explanation

To solve the problem, we need to analyze the properties of periodic and non-periodic functions.

### Step-by-step Solution:

1. **Understanding Periodic Functions**:
A function \( f(x) \) is periodic if there exists a positive constant \( T \) such that:
\[ f(x) = f(x + T) \quad \text{for all } x. \]
Here, \( T \) is called the period of the function.

2. **Understanding Non-Periodic Functions**:
A function \( g(x) \) is non-periodic if there is no such positive constant \( T \) for which \( g(x) = g(x + T) \) holds for all \( x \).

3. **Composition of Functions**:
We need to consider the composition \( g(f(x)) \). Since \( f(x) \) is periodic, it repeats its values after every period \( T \). Therefore, we can write:
\[ f(x + T) = f(x). \]

4. **Evaluating \( g(f(x + T)) \)**:
Since \( f(x + T) = f(x) \), we can substitute this into the function \( g \):
\[ g(f(x + T)) = g(f(x)). \]

5. **Conclusion about \( g(f(x)) \)**:
This shows that:
\[ g(f(x + T)) = g(f(x)). \]
Thus, \( g(f(x)) \) is periodic with the same period \( T \) as \( f(x) \).

6. **Final Statement**:
Therefore, if \( f \) is a periodic function and \( g \) is a non-periodic function, the composition \( g(f(x)) \) is periodic with the period of \( f \).

### Final Answer:
The composition \( g(f(x)) \) is periodic with the same period as \( f(x) \).

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