If f(x)={x}+{2x}` and g(x)=[x]. The number of solutions of f(x)=g(x), where {.} and [x] are respectively the fractional part and greatest functions, is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation \( f(x) = g(x) \) where \( f(x) = \{x\} + \{2x\} \) and \( g(x) = [x] \), we will follow these steps:

### Step 1: Understand the Functions
- The function \( f(x) \) consists of the fractional parts of \( x \) and \( 2x \).
- The function \( g(x) \) is the greatest integer function (floor function) of \( x \).

### Step 2: Set Up the Equation
We need to solve:
\[ \{x\} + \{2x\} = [x] \]

### Step 3: Express the Functions
Recall that:
- \( \{x\} = x - [x] \)
- \( \{2x\} = 2x - [2x] \)

Thus, we can rewrite the equation:
\[ (x - [x]) + (2x - [2x]) = [x] \]
This simplifies to:
\[ 3x - ([x] + [2x]) = [x] \]
Rearranging gives:
\[ 3x = 2[x] + [2x] \]

### Step 4: Analyze the Greatest Integer Function
Let \( [x] = n \) where \( n \) is an integer. Then:
\[ n \leq x < n + 1 \]
This implies:
\[ [x] = n \quad \text{and} \quad [2x] = [2n] \text{ or } [2n + 1] \text{ depending on } x. \]

### Step 5: Case Analysis
1. **Case 1:** If \( n \) is even, then \( [2x] = 2n \).
\[ 3x = 2n + 2n = 4n \implies x = \frac{4n}{3} \]
For \( n \leq x < n + 1 \):
\[ n \leq \frac{4n}{3} < n + 1 \]
This leads to:
\[ 3n \leq 4n \quad \text{(always true)} \]
\[ 4n < 3n + 3 \implies n < 3 \]
Possible values for \( n \) are \( 0, 1, 2 \).

2. **Case 2:** If \( n \) is odd, then \( [2x] = 2n + 1 \).
\[ 3x = 2n + (2n + 1) = 4n + 1 \implies x = \frac{4n + 1}{3} \]
For \( n \leq x < n + 1 \):
\[ n \leq \frac{4n + 1}{3} < n + 1 \]
This leads to:
\[ 3n \leq 4n + 1 \implies n \geq -1 \quad \text{(always true)} \]
\[ 4n + 1 < 3n + 3 \implies n < 2 \]
Possible values for \( n \) are \( 0, 1 \).

### Step 6: Collect Solutions
From both cases, we find:
- For \( n = 0 \): \( x = 0 \)
- For \( n = 1 \): \( x = \frac{5}{3} \)
- For \( n = 2 \): \( x = \frac{8}{3} \)

### Conclusion
The total number of solutions is **3**: \( 0, \frac{5}{3}, \frac{8}{3} \).

### Final Answer
The number of solutions of \( f(x) = g(x) \) is **3**.

---