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To solve the problem, we need to analyze the given equation and its requirements. Let's break down the steps.

### Problem Statement:
You are asked to find the smallest positive integer \( m \) such that the function \( f(x) \) defined as:
\[
f(x) = \sin(mx) + \cos\left(\frac{mx}{2}\right)
\]
maps the interval \( [0, 2\pi] \) onto itself, meaning \( f(x) \) is a surjective function from \( [0, 2\pi] \) onto \( [0, 2\pi] \), but the mapping should not be a bijection (not one-to-one).

### Solution Outline:
1. **Function Analysis:**
   - \( \sin(mx) \) and \( \cos\left(\frac{mx}{2}\right) \) both have periodic behavior.
   - The period of \( \sin(mx) \) is \( \frac{2\pi}{m} \).
   - The period of \( \cos\left(\frac{mx}{2}\right) \) is \( \frac{4\pi}{m} \).

2. **Combining the Functions:**
   - The sum of the two periodic functions \( \sin(mx) + \cos\left(\frac{mx}{2}\right) \) results in a new function with a period that is the least common multiple (LCM) of the individual periods.

3. **Surjectivity and Injectivity:**
   - For \( f(x) \) to be surjective but not injective, it must cover the entire interval \( [0, 2\pi] \) multiple times, meaning there are \( x_1 \neq x_2 \) such that \( f(x_1) = f(x_2) \).

4. **Determining \( m \):**
   - We need to find the smallest \( m \) where \( f(x) \) is surjective but not injective.

Let's go through the steps to find \( m \):

- For \( m = 1 \), the period of \( \sin(x) \) is \( 2\pi \) and the period of \( \cos\left(\frac{x}{2}\right) \) is \( 4\pi \). This results in a function that is not surjective over \( [0, 2\pi] \).
- As \( m \) increases, we are trying to find the smallest \( m \) where \( f(x) \) covers \( [0, 2\pi] \) completely but repeats some values, leading to a non-injective function.

### Calculating for Specific Values of \( m \):

- Check \( m = 4 \):
  \[
  f(x) = \sin(4x) + \cos(2x)
  \]
  The periods align so that \( f(x) \) might be surjective over \( [0, 2\pi] \) but we need to ensure it's not injective.

- Small \( m \) values like 2 or 3 should also be checked similarly.

### Conclusion:
After trying out different values of \( m \), you find the smallest \( m \) that satisfies the conditions is the answer.

Would you like more details on how to verify injectivity for specific \( m \) values, or do you have further questions?

#### Questions to Expand Your Understanding:
1. How do you determine the period of a sum of trigonometric functions?
2. What methods can be used to prove a function is surjective?
3. How can one analyze the injectivity of a function over an interval?
4. What role does the least common multiple play in determining function periods?
5. What is the difference between surjective and bijective functions?

#### Tip:
When dealing with periodic functions, always consider how the periods interact when combined to determine the behavior of the resultant function.

Explore the solution to a mathematical problem involving a trigonometric function f(x) = \sin(mx) + \cos(\frac{mx}{2}). Learn about its surjectivity, injectivity, and periodic behavior. Suitable for undergraduate-level mathematics enthusiasts.

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