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. `[0, 2]`
B. `[-2, 0]`
C. `[-2,2]`
D. `[-2,2]`

Explanation

To solve the problem, we need to find the number of integral points in the range of the composite function \( g(f(x)) \), where the functions \( f(x) \) and \( g(x) \) are defined as follows:

1. \( f(x) = \begin{cases} [x] & , -2 \leq x \leq -1 \\ |x| + 1 & , -1 < x \leq 2 \end{cases} \)

2. \( g(x) = \begin{cases} [x] & , -\pi \leq x < 0 \\ \sin x & , 0 \leq x \leq \pi \end{cases} \)

### Step 1: Determine the range of \( f(x) \)

**For \( -2 \leq x \leq -1 \):**
- Here, \( f(x) = [x] \).
- The greatest integer function \( [x] \) will take values:
- For \( x = -2 \), \( f(-2) = -2 \)
- For \( x = -1 \), \( f(-1) = -1 \)
- Thus, \( f(x) \) will take the values \( -2 \) and \( -1 \).

**For \( -1 < x \leq 2 \):**
- Here, \( f(x) = |x| + 1 \).
- For \( -1 < x < 0 \), \( |x| = -x \) so \( f(x) = -x + 1 \).
- As \( x \) approaches \( -1 \), \( f(x) \) approaches \( 2 \).
- As \( x \) approaches \( 0 \), \( f(x) \) approaches \( 1 \).
- Thus, \( f(x) \) takes values in the interval \( (1, 2) \).

- For \( 0 < x \leq 2 \), \( |x| = x \) so \( f(x) = x + 1 \).
- At \( x = 0 \), \( f(0) = 1 \).
- At \( x = 2 \), \( f(2) = 3 \).
- Thus, \( f(x) \) takes values in the interval \( (1, 3] \).

Combining both parts, the range of \( f(x) \) is:
- From \( -2 \) to \( -1 \) (inclusive) and from \( 1 \) to \( 3 \) (exclusive).

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

Now, we need to evaluate \( g(f(x)) \) based on the range found for \( f(x) \).

**Case 1: When \( f(x) \) is in the range \( [-2, -1] \):**
- \( g(f(x)) = g([-2, -1]) \).
- For \( -2 \leq x < 0 \), \( g(x) = [x] \).
- Thus, \( g(-2) = -2 \) and \( g(-1) = -1 \).
- Therefore, \( g(f(x)) \) takes values \( -2 \) and \( -1 \).

**Case 2: When \( f(x) \) is in the range \( (1, 3] \):**
- \( g(f(x)) = g(f(x)) = \sin(f(x)) \).
- The sine function varies between \( 0 \) and \( 1 \) for \( x \) in the interval \( [0, \pi] \).
- Therefore, \( g(f(x)) \) will take values in the interval \( (0, 1] \).

### Step 3: Combine the ranges

Combining the ranges from both cases:
- From \( g(f(x)) \), we have:
- Values from case 1: \( -2, -1 \)
- Values from case 2: \( (0, 1] \)

### Step 4: Identify integral points in the combined range

The integral points in the combined range \( \{-2, -1\} \cup (0, 1] \) are:
- From \( \{-2, -1\} \): \( -2, -1 \)
- From \( (0, 1] \): The only integral point is \( 1 \).

Thus, the integral points are \( -2, -1, 1 \).

### Final Count of Integral Points

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