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 \( \{x\} \) takes values in the interval \([0, 1)\) for any real number \( x \).

### Step 2: Analyze the function \( y = 2^{\{x\}} \)
Since \( \{x\} \) varies from \( 0 \) to just below \( 1 \), we can substitute this into our function:
- When \( \{x\} = 0 \), we have:
\[ y = 2^0 = 1 \]
- As \( \{x\} \) approaches \( 1 \) (but never actually reaches it), we have:
\[ y = 2^{0.999...} \text{ (which approaches 2)} \]

### Step 3: Determine the range of \( y \)
From the analysis:
- The minimum value of \( y \) occurs when \( \{x\} = 0 \), which gives \( y = 1 \).
- The maximum value of \( y \) occurs as \( \{x\} \) approaches \( 1 \), which gives \( y \) approaching \( 2 \) but never reaching it.

Thus, the range of \( y = 2^{\{x\}} \) is:
\[ [1, 2) \]

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