Let `f (x)= {{:(x [x] , 0 le x lt 2),( (x-1), 2 le x le 3):}` where [x]= greatest integer less than or equal to x, then:
The number of integers in the range of `y =f (x)` is:

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined in two parts:

1. For \( 0 \leq x < 2 \): \( f(x) = x \cdot [x] \)
2. For \( 2 \leq x \leq 3 \): \( f(x) = x - 1 \)

Where \( [x] \) is the greatest integer function (also known as the floor function).

### Step 1: Analyze \( f(x) \) for \( 0 \leq x < 2 \)

In this interval, we need to determine the values of \( f(x) \):

- For \( 0 \leq x < 1 \):
- Here, \( [x] = 0 \)
- Thus, \( f(x) = x \cdot 0 = 0 \)

- For \( 1 \leq x < 2 \):
- Here, \( [x] = 1 \)
- Thus, \( f(x) = x \cdot 1 = x \)

So, in the interval \( 0 \leq x < 2 \), the function takes the values:
- \( f(x) = 0 \) when \( 0 \leq x < 1 \)
- \( f(x) = x \) when \( 1 \leq x < 2 \)

This means \( f(x) \) ranges from \( 0 \) to just below \( 2 \) (not including \( 2 \)).

### Step 2: Analyze \( f(x) \) for \( 2 \leq x \leq 3 \)

In this interval, we calculate \( f(x) \):

- For \( 2 \leq x \leq 3 \):
- Here, \( f(x) = x - 1 \)

Evaluating the endpoints:
- At \( x = 2 \): \( f(2) = 2 - 1 = 1 \)
- At \( x = 3 \): \( f(3) = 3 - 1 = 2 \)

Thus, in the interval \( 2 \leq x \leq 3 \), the function takes values from \( 1 \) to \( 2 \) (including both).

### Step 3: Combine the ranges from both intervals

Now we combine the ranges from both parts of the function:
- From \( 0 \leq x < 2 \): The range is \( [0, 2) \) which includes \( 0 \) and all values up to but not including \( 2 \).
- From \( 2 \leq x \leq 3 \): The range is \( [1, 2] \) which includes \( 1 \) and \( 2 \).

### Step 4: Determine the integers in the combined range

The combined range of \( f(x) \) is:
- From \( 0 \) to just below \( 2 \) (from the first part)
- From \( 1 \) to \( 2 \) (from the second part)

The integers in this combined range are:
- \( 0 \)
- \( 1 \)
- \( 2 \)

### Conclusion

Thus, the integers in the range of \( f(x) \) are \( 0, 1, 2 \). Therefore, the number of integers in the range of \( y = f(x) \) is **3**.

### Final Answer
The number of integers in the range of \( y = f(x) \) is **3**.