Consider two functions
`f(x)={([x]",",-2le x le -1),(|x|+1",",-1 lt x le 2):} and g(x)={([x]",",-pi le x lt 0),(sinx",",0le x le pi):}`
where [.] denotes the greatest integer function.
The number of integral points in the range of `g(f(x))` is

Options

A. 2
B. 4
C. 3
D. 5

Explanation

To solve the problem, we need to find the number of integral points in the range of the composite function \( g(f(x)) \). Let's break this down step by step.

### Step 1: Define the functions \( f(x) \) and \( g(x) \)

The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} x & \text{for } -2 \leq x \leq -1 \\ |x| + 1 & \text{for } -1 < x \leq 2 \end{cases} \]

The function \( g(x) \) is defined as:
\[ g(x) = \begin{cases} x & \text{for } -\pi \leq x < 0 \\ \sin x & \text{for } 0 \leq x \leq \pi \end{cases} \]

### Step 2: Determine the range of \( f(x) \)

1. For \( -2 \leq x \leq -1 \):
- \( f(x) = x \)
- The range of \( f(x) \) in this interval is \( [-2, -1] \).

2. For \( -1 < x \leq 2 \):
- \( f(x) = |x| + 1 = x + 1 \) (since \( x \) is positive in this interval)
- When \( x = -1 \), \( f(-1) = 0 \).
- When \( x = 2 \), \( f(2) = 3 \).
- The range of \( f(x) \) in this interval is \( (0, 3] \).

Combining both intervals, the overall range of \( f(x) \) is:
\[ [-2, -1] \cup (0, 3] \]

### Step 3: Determine the range of \( g(f(x)) \)

Now we need to find \( g(f(x)) \) based on the range of \( f(x) \).

1. For \( f(x) \) values in \( [-2, -1] \):
- Since \( -2 \leq f(x) < -1 \), we use \( g(x) = x \).
- Thus, \( g(f(x)) = f(x) \) results in the range \( [-2, -1] \).

2. For \( f(x) \) values in \( (0, 3] \):
- Since \( 0 < f(x) \leq 3 \), we use \( g(x) = \sin x \).
- We need to find \( \sin(f(x)) \) for \( f(x) \) in \( (0, 3] \).

### Step 4: Calculate \( \sin(f(x)) \) for \( f(x) \in (0, 3] \)

- The sine function oscillates between 0 and 1 for \( x \) in \( (0, \pi) \) and then decreases from 1 to 0 as \( x \) approaches \( \pi \).
- For \( f(x) \) in \( (0, 3] \):
- \( \sin(f(x)) \) will take values from \( \sin(0) = 0 \) to \( \sin(3) \).

### Step 5: Determine the range of \( g(f(x)) \)

Combining the ranges:
- From \( [-2, -1] \), we have values \( -2 \) and \( -1 \).
- From \( (0, 3] \), we have values from \( 0 \) to \( \sin(3) \).

### Step 6: Identify integral points in the range

1. From \( [-2, -1] \), the integral points are \( -2, -1 \).
2. From \( (0, \sin(3)] \):
- Since \( \sin(3) \) is approximately \( 0.1411 \), the only integral point in this range is \( 0 \).

### Final Count of Integral Points

The integral points from both ranges are:
- From \( [-2, -1] \): \( -2, -1 \)
- From \( (0, \sin(3)] \): \( 0 \)

Thus, the total number of integral points in the range of \( g(f(x)) \) is:
\[ \text{Total Integral Points} = 3 \]

### Summary

The number of integral points in the range of \( g(f(x)) \) is **3**.