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.