Let f(x) = `x[x] , x cancelin Z ` [.] denotes greatest integer function), then f (x ) is equal to

Options

A. 2x
B. [x]
C. 2 [x]
D. none of these

Answer

To solve the problem, we need to analyze the function \( f(x) = x \cdot [x] \), where \([x]\) denotes the greatest integer function (also known as the floor function). This function is defined for \( x \) not being an integer.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \( x \). For example:
- If \( x = 0.1 \), then \([0.1] = 0\)
- If \( x = 1.3 \), then \([1.3] = 1\)
- If \( x = 2.7 \), then \([2.7] = 2\)

2. **Evaluating \( f(x) \) for Different Intervals**:
We will evaluate \( f(x) \) for different ranges of \( x \):

- **For \( 0 < x < 1 \)**:
\[ f(x) = x \cdot [x] = x \cdot 0 = 0 \]

- **For \( 1 < x < 2 \)**:
\[ f(x) = x \cdot [x] = x \cdot 1 = x \]

- **For \( 2 < x < 3 \)**:
\[ f(x) = x \cdot [x] = x \cdot 2 = 2x \]

3. **Summarizing the Function**:
From the evaluations, we can summarize \( f(x) \) as follows:
- \( f(x) = 0 \) for \( 0 < x < 1 \)
- \( f(x) = x \) for \( 1 < x < 2 \)
- \( f(x) = 2x \) for \( 2 < x < 3 \)

4. **Graphing the Function**:
- For \( 0 < x < 1 \), the graph is a horizontal line at \( y = 0 \).
- For \( 1 < x < 2 \), the graph is a line with a slope of 1 (i.e., \( y = x \)).
- For \( 2 < x < 3 \), the graph is a line with a slope of 2 (i.e., \( y = 2x \)).

5. **Conclusion**:
The function \( f(x) \) is piecewise defined and varies based on the interval in which \( x \) lies. The function does not have a single expression that describes it for all \( x \), but it can be represented as:
\[ f(x) = \begin{cases} 0 & \text{for } 0 < x < 1 \\ x & \text{for } 1 < x < 2 \\ 2x & \text{for } 2 < x < 3 \end{cases} \]