If two functions f(x) and g(x) are commutative with period p, show that the function af(x) + bg(x) with any fixed number a, b is also periodic with period p 

To show that the function \( h(x) = a f(x) + b g(x) \) is periodic with period \( p \), where \( f(x) \) and \( g(x) \) are commutative functions with period \( p \), we proceed as follows:

### Given:
- \( f(x) \) and \( g(x) \) are periodic with period \( p \), which means:
  \[
  f(x + p) = f(x) \quad \text{and} \quad g(x + p) = g(x) \quad \forall x
  \]
- \( h(x) = a f(x) + b g(x) \), where \( a \) and \( b \) are constants.

### To show:
The function \( h(x) \) is periodic with period \( p \), i.e., \( h(x + p) = h(x) \).

### Proof:
1.

If two functions f(x) and g(x) are commutative with period p, show that the function af(x) + bg(x) with any fixed number a, b is also periodic with period p.

Understand how to prove that the function af(x) + bg(x) is periodic with period p, given that f(x) and g(x) are commutative and periodic. This solution explores key mathematical concepts like periodicity, commutativity, and function addition, suitable for advanced high school or early university students.

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