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 \([x]\) is the greatest integer function, we will follow these steps:

### Step 1: Rewrite the Function
We can simplify the function as follows:
\[ f(x) = \frac{2^x}{2^{[x]}} = 2^{x - [x]} \]
Here, \(x - [x]\) represents the fractional part of \(x\), denoted as \(\{x\}\). Thus, we can rewrite the function as:
\[ f(x) = 2^{\{x\}} \]

### Step 2: Determine the Domain
The domain of \(f(x)\) is the set of all real numbers since there are no restrictions on \(x\) for the function to be defined. Therefore, the domain is:
\[ \text{Domain} = \mathbb{R} \]

### Step 3: Determine the Range
The fractional part \(\{x\}\) varies between 0 (inclusive) and 1 (exclusive). Therefore, we have:
\[ 0 \leq \{x\} < 1 \]
Now, substituting this into the function:
- When \(\{x\} = 0\), \(f(x) = 2^0 = 1\).
- As \(\{x\}\) approaches 1, \(f(x)\) approaches \(2^1 = 2\) but never reaches it.

Thus, the range of \(f(x)\) is:
\[ \text{Range} = [1, 2) \]

### Step 4: Graph the Function
To graph the function, we note that:
- At every integer \(n\), \(f(n) = 2^{\{n\}} = 2^0 = 1\).
- As \(x\) increases from \(n\) to \(n+1\) (where \(n\) is an integer), \(f(x)\) increases from 1 to just below 2.

The graph will consist of segments that start at \(y = 1\) at each integer and rise exponentially towards \(y = 2\) but never actually reaching it. The graph will be periodic with a repeating pattern every integer.

### Final Summary
- **Domain**: \(\mathbb{R}\) (all real numbers)
- **Range**: \([1, 2)\) (1 is included, 2 is not included)
- **Graph**: The graph will have horizontal lines at \(y=1\) at each integer \(x\) and will rise towards \(y=2\) as \(x\) approaches the next integer.