Find the range of `f(x) = e^(x-[x]), x in R`

Answer

To find the range of the function \( f(x) = e^{(x - [x])} \), where \( [x] \) is the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Understand the function
The function \( f(x) = e^{(x - [x])} \) can be rewritten using the definition of the greatest integer function. The term \( x - [x] \) represents the fractional part of \( x \), denoted as \( \{x\} \). Therefore, we can express the function as:
\[ f(x) = e^{\{x\}} \]
where \( \{x\} = x - [x] \) and \( 0 \leq \{x\} < 1 \).

### Step 2: Determine the range of the fractional part
Since the fractional part \( \{x\} \) takes values in the interval \([0, 1)\), we can analyze how this affects \( f(x) \):
\[ 0 \leq \{x\} < 1 \]

### Step 3: Find the range of \( e^{\{x\}} \)
Now we need to find the range of \( f(x) = e^{\{x\}} \). Since \( e^y \) is an increasing function, we can evaluate the endpoints of the interval for \( \{x\} \):
- When \( \{x\} = 0 \), \( f(x) = e^0 = 1 \).
- As \( \{x\} \) approaches 1 (but does not include it), \( f(x) \) approaches \( e^1 = e \).

### Step 4: Conclude the range
Thus, the range of \( f(x) \) is:
\[ [1, e) \]

### Final Answer
The range of the function \( f(x) = e^{(x - [x])} \) is \([1, e)\).

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