Draw the graph of `f(x)=(2^(x))/(2^([x]))` where `[.]` represents greatest integer function and find the domain and range.

Answer

To solve the problem of drawing the graph of the function \( f(x) = \frac{2^x}{2^{[x]}} \) where \([x]\) represents the greatest integer function, we will follow these steps:

### Step 1: Simplify the Function
The function can be rewritten as:
\[ f(x) = \frac{2^x}{2^{[x]}} = 2^{x - [x]} \]
Here, \(x - [x]\) is the fractional part of \(x\), denoted as \(\{x\}\). Therefore, we can express the function as:
\[ f(x) = 2^{\{x\}} \]

### Step 2: Analyze the Fractional Part
The fractional part \(\{x\}\) takes values in the interval \([0, 1)\). This means that \(f(x)\) will take values based on the exponent \(2^{\{x\}}\) where \(\{x\}\) varies from 0 to just below 1.

### Step 3: Determine Values of the Function
- When \(x\) is an integer (e.g., \(x = 0, 1, 2, \ldots\)), \(\{x\} = 0\), so:
\[ f(x) = 2^0 = 1 \]
- When \(x\) is not an integer, \(\{x\}\) will be in the range \((0, 1)\). Therefore, \(f(x)\) will take values in the interval:
\[ f(x) = 2^{\{x\}} \text{ where } \{x\} \in [0, 1) \implies f(x) \in [1, 2) \]

### Step 4: Draw the Graph
1. For every integer \(n\), the function \(f(x)\) equals 1 at \(x = n\).
2. As \(x\) approaches the next integer \(n+1\) from the left, \(f(x)\) approaches \(2\) but never reaches it.
3. The graph will consist of segments that rise from \(1\) to just below \(2\) for each interval \([n, n+1)\).

### Step 5: Identify the Domain and Range
- **Domain**: The function is defined for all real numbers \(x\):
\[ \text{Domain} = \mathbb{R} \]
- **Range**: The function takes values from \(1\) to just below \(2\):
\[ \text{Range} = [1, 2) \]

### Final Graph
The graph will consist of horizontal line segments at \(y = 1\) for each integer \(x\) and will rise to just below \(2\) as \(x\) approaches the next integer.