The set of all values of x satisfying `{x}=x[xx] " where " [xx]` represents greatest integer function `{xx}` represents fractional part of x

Options

A. 0
B. `-(1)/(2)`
C. `-1 lt x lt 1`
D. Both A and B

Explanation

To solve the equation \(\{x\} = x [x x]\), where \([x]\) represents the greatest integer function and \(\{x\}\) represents the fractional part of \(x\), we will follow these steps:

### Step 1: Understand the Definitions
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part \(\{x\}\) is defined as \(x - [x]\).

### Step 2: Rewrite the Equation
Given the equation:
\[ \{x\} = x [x x] \]
We can express \(\{x\}\) as:
\[ \{x\} = x - [x] \]
Thus, we rewrite the equation as:
\[ x - [x] = x [x x] \]

### Step 3: Analyze the Right Side
The term \([x x]\) is the greatest integer of \(x^2\). Therefore, we can rewrite the equation as:
\[ x - [x] = x [x^2] \]

### Step 4: Set Up the Inequalities
Since \(\{x\}\) must lie between 0 and 1, we have:
\[ 0 \leq x - [x] < 1 \]
This implies:
\[ 0 \leq x [x^2] < 1 \]

### Step 5: Analyze the Denominator
To ensure that the denominator in the expression \(\frac{x^2}{1 + x}\) is greater than zero, we need:
\[ 1 + x > 0 \implies x > -1 \]

### Step 6: Solve the Quadratic Inequality
Next, we need to solve the inequality:
\[ x^2 - x - 1 < 0 \]
To find the roots of the quadratic equation \(x^2 - x - 1 = 0\), we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{5}}{2} \]
The roots are:
\[ x_1 = \frac{1 + \sqrt{5}}{2}, \quad x_2 = \frac{1 - \sqrt{5}}{2} \]

### Step 7: Determine the Interval
The quadratic \(x^2 - x - 1\) is negative between its roots:
\[ \frac{1 - \sqrt{5}}{2} < x < \frac{1 + \sqrt{5}}{2} \]

### Step 8: Combine Conditions
We also have the condition \(x > -1\). Thus, we need to find the intersection of the intervals:
- From the quadratic inequality: \(\left(\frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2}\right)\)
- From the condition \(x > -1\).

### Step 9: Final Solution
The final solution set for \(x\) is:
\[ x \in \left(\frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2}\right) \]