The period of function `2^({x}) +sin pi x+3^({x//2})+cos pi x` (where {x} denotes the fractional part of x) is

Options

A. 2
B. 1
C. 3
D. None of these

Explanation

To find the period of the function \( f(x) = 2^{\{x\}} + \sin(\pi x) + 3^{\frac{\{x\}}{2}} + \cos(\pi x) \), where \(\{x\}\) denotes the fractional part of \(x\), we will analyze the periods of each component of the function.

### Step 1: Determine the period of \(2^{\{x\}}\)
The fractional part function \(\{x\}\) has a period of 1, meaning \(\{x + 1\} = \{x\}\). Therefore, the function \(2^{\{x\}}\) also has a period of 1.

**Hint:** The period of a function that depends on the fractional part of \(x\) is typically the same as the period of the fractional part itself.

### Step 2: Determine the period of \(\sin(\pi x)\)
The sine function \(\sin(kx)\) has a period of \(\frac{2\pi}{k}\). Here, \(k = \pi\), so the period of \(\sin(\pi x)\) is:
\[ \text{Period} = \frac{2\pi}{\pi} = 2 \]

**Hint:** For sine functions, remember to use the formula for the period based on the coefficient of \(x\).

### Step 3: Determine the period of \(3^{\frac{\{x\}}{2}}\)
Similar to \(2^{\{x\}}\), the function \(3^{\frac{\{x\}}{2}}\) depends on \(\{x\}\), which has a period of 1. Thus, the period of \(3^{\frac{\{x\}}{2}}\) is also 1.

**Hint:** Exponential functions with a variable in the exponent that depends on the fractional part will share the same period as the fractional part.

### Step 4: Determine the period of \(\cos(\pi x)\)
The cosine function \(\cos(kx)\) also has a period of \(\frac{2\pi}{k}\). For \(\cos(\pi x)\), we have:
\[ \text{Period} = \frac{2\pi}{\pi} = 2 \]

**Hint:** Just like sine, cosine functions have periods based on the coefficient of \(x\).

### Step 5: Find the least common multiple (LCM) of the periods
Now we have determined the periods of each component:
- Period of \(2^{\{x\}} = 1\)
- Period of \(\sin(\pi x) = 2\)
- Period of \(3^{\frac{\{x\}}{2}} = 1\)
- Period of \(\cos(\pi x) = 2\)

The periods are \(1, 2, 1, 2\). The least common multiple of these periods is:
\[ \text{LCM}(1, 2) = 2 \]

### Conclusion
Thus, the period of the function \( f(x) = 2^{\{x\}} + \sin(\pi x) + 3^{\frac{\{x\}}{2}} + \cos(\pi x) \) is \( \boxed{2} \).

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