Find the fundamental period of the following function:
`f(x)=2+3cos(x-2)`

Explanation

To find the fundamental period of the function \( f(x) = 2 + 3 \cos(x - 2) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Basic Function**:
The function \( f(x) \) is based on the cosine function, which is \( \cos(x) \). The general form of the cosine function is \( \cos(kx) \), where \( k \) affects the period of the function.

2. **Determine the Period of the Cosine Function**:
The standard period of the cosine function \( \cos(x) \) is \( 2\pi \). This means that the function repeats every \( 2\pi \) units.

3. **Analyze the Transformation**:
In the function \( f(x) = 2 + 3 \cos(x - 2) \), the term \( (x - 2) \) indicates a horizontal shift to the right by 2 units. However, this does not affect the period of the function.

4. **Consider the Amplitude and Vertical Shift**:
The coefficient \( 3 \) in front of the cosine function affects the amplitude (the height of the peaks and depth of the troughs) but does not affect the period. The addition of \( 2 \) shifts the entire graph upwards by 2 units, which also does not affect the period.

5. **Conclusion**:
Since the transformations applied (horizontal shift, vertical shift, and amplitude change) do not affect the period of the cosine function, the fundamental period of \( f(x) = 2 + 3 \cos(x - 2) \) remains the same as that of \( \cos(x) \).

Thus, the fundamental period of the function is:
\[ \text{Period} = 2\pi \]

### Final Answer:
The fundamental period of the function \( f(x) = 2 + 3 \cos(x - 2) \) is \( 2\pi \).

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