find boundaries for function f(x)=4sin(x)+cos(2x)

To find the boundaries (or range) of the function \( f(x) = 4\sin(x) + \cos(2x) \), let's break it down step by step:

1. **Rewriting the terms**:
   - \( \sin(x) \) oscillates between -1 and 1.
   - \( \cos(2x) \) also oscillates between -1 and 1, but with a double frequency.

2. **Analyze the range of individual terms**:
   - The term \( 4\sin(x) \) ranges between \( 4(-1) = -4 \) and \( 4(1) = 4 \).
   - The term \( \cos(2x) \) ranges between -1 and 1.

3. **Considering their sum**:
   The function is a sum of \( 4\sin(x) \) and \( \cos(2x) \). To find the overall range, we analyze the maximum and minimum values of their sum by considering different combinations of their individual maxima and minima.

4. **Maximizing and minimizing the sum**:
   - **Maximum value**: When \( \sin(x) = 1 \) and \( \cos(2x) = 1 \), the function is \( 4(1) + 1 = 5 \).
   - **Minimum value**: When \( \sin(x) = -1 \) and \( \cos(2x) = -1 \), the function is \( 4(-1) + (-1) = -5 \).

Thus, the function \( f(x) = 4\sin(x) + \cos(2x) \) has a range from \( -5 \) to \( 5 \).

### Conclusion:
The boundaries (range) of \( f(x) = 4\sin(x) + \cos(2x) \) are:
\[
[-5, 5]
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the range of \( g(x) = 3\sin(x) + 2\cos(x) \)?
2. How does the range of \( f(x) = \sin(x) + \cos(2x) \) differ from the original function?
3. How do frequency and amplitude of trigonometric functions affect their range?
4. What is the graphical representation of \( f(x) = 4\sin(x) + \cos(2x) \)?
5. How would you find the boundaries of \( f(x) = a\sin(x) + b\cos(2x) \)?

**Tip**: The sum of sine and cosine functions often leads to periodic functions. Their maxima and minima depend on both amplitude and phase shift.

Explore how to find the boundaries (range) of the function f(x) = 4sin(x) + cos(2x). This solution walks through the oscillatory behavior of trigonometric functions and how to calculate their range, suitable for high school students in Grades 10-12.

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