To find the range of the function \( f(x) = \frac{x \lfloor x \rfloor}{1+x} \) defined on the interval \( (1, 3) \), we will analyze the function in two parts based on the behavior of the greatest integer function \( \lfloor x \rfloor \).
### Step 1: Split the Interval
The interval \( (1, 3) \) can be split into two subintervals:
1. \( (1, 2) \)
2. \( [2, 3) \)
### Step 2: Analyze the First Interval \( (1, 2) \)
In the interval \( (1, 2) \):
- The greatest integer function \( \lfloor x \rfloor = 1 \) for all \( x \) in this interval.
- Therefore, the function simplifies to:
\[
f(x) = \frac{x \cdot 1}{1+x} = \frac{x}{1+x}
\]
### Step 3: Find the Range for \( (1, 2) \)
To find the range of \( f(x) = \frac{x}{1+x} \) as \( x \) varies from 1 to 2:
- Calculate \( f(1) \):
\[
f(1) = \frac{1}{1+1} = \frac{1}{2}
\]
- Calculate \( f(2) \):
\[
f(2) = \frac{2}{1+2} = \frac{2}{3}
\]
- Since \( f(x) \) is continuous and increasing in \( (1, 2) \), the range for this interval is:
\[
\left( \frac{1}{2}, \frac{2}{3} \right)
\]
### Step 4: Analyze the Second Interval \( [2, 3) \)
In the interval \( [2, 3) \):
- The greatest integer function \( \lfloor x \rfloor = 2 \) for all \( x \) in this interval.
- Therefore, the function simplifies to:
\[
f(x) = \frac{x \cdot 2}{1+x} = \frac{2x}{1+x}
\]
### Step 5: Find the Range for \( [2, 3) \)
To find the range of \( f(x) = \frac{2x}{1+x} \) as \( x \) varies from 2 to 3:
- Calculate \( f(2) \):
\[
f(2) = \frac{2 \cdot 2}{1+2} = \frac{4}{3}
\]
- Calculate \( f(3) \) (approaching from the left):
\[
f(3) = \frac{2 \cdot 3}{1+3} = \frac{6}{4} = \frac{3}{2}
\]
- Since \( f(x) \) is continuous and increasing in \( [2, 3) \), the range for this interval is:
\[
\left[ \frac{4}{3}, \frac{3}{2} \right)
\]
### Step 6: Combine the Ranges
Now, we combine the ranges from both intervals:
1. From \( (1, 2) \): \( \left( \frac{1}{2}, \frac{2}{3} \right) \)
2. From \( [2, 3) \): \( \left[ \frac{4}{3}, \frac{3}{2} \right) \)
### Final Range
Thus, the overall range of \( f(x) \) on the interval \( (1, 3) \) is:
\[
\left( \frac{1}{2}, \frac{2}{3} \right) \cup \left[ \frac{4}{3}, \frac{3}{2} \right)
\]
