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 and usr Linear combination of two functions

To show that the function \( h(x) = af(x) + bg(x) \), where \( a \) and \( b \) are constants, is also periodic with period \( p \) if \( f(x) \) and \( g(x) \) are periodic with period \( p \), we need to:

1. Use the fact that a function \( f(x) \) is periodic with period \( p \) if and only if \( f(x + p) = f(x) \) for all \( x \).
2. Apply the linear combination of the two functions \( f(x) \) and \( g(x) \) to prove that \( h(x + p) = h(x) \), implying periodicity with period \( p \).

### Given:
- \( f(x + p) = f(x) \) and \( g(x + p) = g(x) \), meaning \( f(x) \) and \( g(x) \) are periodic with period \( p \).
- The function \( h(x) = af(x) + bg(x) \) is a linear combination of \( f(x) \) and \( g(x) \), where \( a \) and \( b \) are constants.

### Proof:

We want to show that \( h(x) = af(x) + bg(x) \) is periodic with period \( p \). That is, we need to show:

\[
h(x + p) = h(x).
\]

#### Step 1: Evaluate \( h(x + p) \).

\[
h(x + p) = a f(x + p) + b g(x + p).
\]

#### Step 2: Use the periodicity of \( f(x) \) and \( g(x) \).

Since \( f(x) \) and \( g(x) \) are periodic with period \( p \), we know that:

\[
f(x + p) = f(x) \quad \text{and} \quad g(x + p) = g(x).
\]

Thus, we can substitute into the expression for \( h(x + p) \):

\[
h(x + p) = a f(x) + b g(x).
\]

#### Step 3: Compare with \( h(x) \).

By definition, \( h(x) = af(x) + bg(x) \). Therefore, we have:

\[
h(x + p) = h(x).
\]

This shows that \( h(x) = af(x) + bg(x) \) is periodic with period \( p \).

### Conclusion:

We have proven that the function \( h(x) = af(x) + bg(x) \) is periodic with period \( p \), given that \( f(x) \) and \( g(x) \) are periodic with the same period \( p \).

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Would you like more details on this, or do you have any further questions?

Here are 5 related questions you might find interesting:
1. What is the general form of a periodic function?
2. How does the commutative property of periodic functions affect their combination?
3. Can the period of a linear combination of functions change based on different weights \(a\) and \(b\)?
4. How do you prove that the sum of two periodic functions with different periods is not necessarily periodic?
5. What are some real-world examples of periodic functions in physics?

**Tip:** When working with periodic functions, always check their periods before combining them, as their periods must match for the result to be periodic.

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.

This solution demonstrates how to prove that the linear combination of two commutative functions, af(x) + bg(x), is also periodic with period p. The proof uses fundamental properties of periodicity and the commutative property of functions. Suitable for undergraduate mathematics students.

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