Period of the function `f(x) = cos(cospix) +e^({4x})`, where `{.}` denotes the fractional part of x, is

Answer

To find the period of the function \( f(x) = \cos(\cos(\pi x)) + e^{\{4x\}} \), where \( \{x\} \) denotes the fractional part of \( x \), we will analyze the periodic components of the function step by step.

### Step 1: Identify the periodic components
The function consists of two parts:
1. \( \cos(\cos(\pi x)) \)
2. \( e^{\{4x\}} \)

### Step 2: Determine the period of \( \cos(\cos(\pi x)) \)
The inner function \( \cos(\pi x) \) has a period of 2 because:
- The cosine function \( \cos(x) \) has a period of \( 2\pi \).
- Therefore, \( \cos(\pi x) \) has a period of \( \frac{2\pi}{\pi} = 2 \).

Now, since \( \cos(x) \) is periodic with a period of \( 2\pi \), and we are taking the cosine of \( \cos(\pi x) \), the period remains 2. Thus, the period of \( \cos(\cos(\pi x)) \) is also 2.

### Step 3: Determine the period of \( e^{\{4x\}} \)
The fractional part function \( \{x\} \) has a period of 1. Therefore, \( \{4x\} \) also has a period of \( \frac{1}{4} \) because:
- The function \( \{kx\} \) has a period of \( \frac{1}{k} \).

Thus, the period of \( e^{\{4x\}} \) is 1.

### Step 4: Find the overall period of \( f(x) \)
To find the overall period of \( f(x) \), we need to find the least common multiple (LCM) of the two periods we found:
- Period of \( \cos(\cos(\pi x)) \) is 2 (denote this as \( T_1 \)).
- Period of \( e^{\{4x\}} \) is \( \frac{1}{4} \) (denote this as \( T_2 \)).

### Step 5: Calculate the LCM
To find the LCM of \( T_1 = 2 \) and \( T_2 = \frac{1}{4} \):
- Convert \( T_1 \) into a fraction: \( T_1 = \frac{8}{4} \).
- Now we find the LCM of \( \frac{8}{4} \) and \( \frac{1}{4} \):
- The LCM of the numerators (8 and 1) is 8.
- The denominator remains 4.

Thus, the LCM is \( \frac{8}{4} = 2 \).

### Conclusion
The period of the function \( f(x) = \cos(\cos(\pi x)) + e^{\{4x\}} \) is \( 2 \).