The range of `y=2^({x})` is , where `{.}` fractional part function

Options

A. `(0,1]`
B. `[1,2]`
C. `(0,oo)`
D. none of these

Answer

To find the range of the function \( y = 2^{\{x\}} \), where \( \{x\} \) denotes the fractional part of \( x \), we can follow these steps:

### Step 1: Understand the fractional part function
The fractional part function \( \{x\} \) is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
This means that \( \{x\} \) gives us the decimal part of \( x \), which always lies in the interval:
\[ 0 \leq \{x\} < 1 \]

### Step 2: Analyze the range of \( \{x\} \)
From the definition, we can conclude that the range of \( \{x\} \) is:
\[ [0, 1) \]

### Step 3: Substitute the range into the function
Now, we substitute the range of \( \{x\} \) into the function \( y = 2^{\{x\}} \). Since \( \{x\} \) takes values from 0 to just below 1, we can analyze how this affects \( y \):
- When \( \{x\} = 0 \):
\[ y = 2^0 = 1 \]
- As \( \{x\} \) approaches 1 (but does not include 1):
\[ y = 2^{\{x\}} \to 2^1 = 2 \]

### Step 4: Determine the range of \( y \)
Thus, as \( \{x\} \) varies from 0 to just below 1, \( y \) will vary from:
\[ 1 \text{ to } 2 \]
This gives us the range:
\[ [1, 2) \]

### Conclusion
The range of \( y = 2^{\{x\}} \) is:
\[ [1, 2) \]