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When analyzing the cosine function, it's essential to understand the concept of periodicity. Periodicity refers to the characteristic of a function to repeat its values at regular intervals, called the period. For the standard cosine function, \( \cos(x) \), the period is \( 2\pi \), meaning that the function's values repeat every \( 2\pi \) radians.

This periodic nature implies that for any trigonometric function like \( \cos(x) \) and \( \cos(2x) \), understanding one complete cycle from \( 0 \) to \( 2\pi \) is sufficient to know the behavior over all real numbers. Furthermore, the maximum and minimum values repeat with this period as well. This is particularly useful when analyzing a composite function involving cosine terms, as we can limit our investigation to a single period without loss of generality.

In the case of the function \( f(x)=3+4\cos(x)+\cos(2x) \), it combines two periodic cosine functions with different frequencies—\cos(x) repeating every \( 2\pi \) and \cos(2x) repeating every \( \pi \). However, for calculative simplicity, we focus on the common period of \( 2\pi \), within which the combined behavior of these terms can be predicted, and any conclusions drawn will hold true for all values of \( x \) due to periodicity.

The range of a function represents the set of all possible output values it can produce. For trigonometric functions, these ranges are bounded and consistent across their respective periods. The standard cosine and sine functions, for instance, have a range of \( [-1, 1] \), which means they oscillate between \(-1\) and \(+1\).

It's critical to understand that the range doesn't change when the function is transformed, such as multiplying it by a constant or adding a constant to it. For the function \( f(x) = 3 + 4\cos(x) + \cos(2x) \), the range is affected by these transformations. The \( \cos(x) \) term, when multiplied by 4, has a new range of \( [-4, 4] \), and adding 3 shifts all of these values up by 3. Thus, each term contributes to the overall range of \( f(x) \) by its transformed amount.

How Does This Affect Our Function?

In the step by step solution, we saw the worst case when both cosine terms are at their minimum, yielding \( f(x) = 3 - 4 - 1 = -2 \) as the function's minimum possible value. This calculation uses the range concept since by taking the lower bounds of these individual trigonometric functions, we found the overall function's minimum. Hence, the range of \( f(x) \) does include negative numbers.

Determining the sign of a function at various points in its domain is often crucial for understanding its behavior, such as identifying where it crosses the x-axis or when it is above or below a certain value. To determine the sign of a trigonometric function, one must consider the range and periodicity, as well as any transformations applied to the function, like scaling and shifting.

In our example function \( f(x) = 3 + 4\cos(x) + \cos(2x) \), we know that the cosine function fluctuates between -1 and 1. Multiplying by 4 and adding 3 will change the amplitude and baseline of the oscillation but not the fundamental behavior of the cosine wave. By analyzing the minimum and maximum values each term can take, we see that it's possible for \( f(x) \) to be negative if the sum of these values is less than zero.

As shown in the solution, the function reaches a minimum value of -2 when both cosine terms are at their minimum, proving that \( f(x) \) can indeed be negative. To determine the function's sign in general, one could sketch the graph using these principles or analyze it piece by piece, taking into consideration how each transformation affects the overall output.