Define fog(x) and gof(x). Also find their domain and range. (i) `f(x) = [x], g(x) = sin x`

Explanation

To solve the problem, we need to define the compositions of the functions \( f(x) \) and \( g(x) \), and then find their domains and ranges.

### Given Functions:
1. \( f(x) = [x] \) (the greatest integer function)
2. \( g(x) = \sin x \)

### Step 1: Define \( f(g(x)) \) or \( f \circ g(x) \)
To find \( f(g(x)) \):
- Substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(\sin x) = [\sin x] \]
This means we take the greatest integer less than or equal to \( \sin x \).

### Step 2: Find the Domain of \( f(g(x)) \)
- Since \( g(x) = \sin x \) is defined for all real numbers \( x \), and \( f(x) \) is also defined for all real numbers, the domain of \( f(g(x)) \) is:
\[ \text{Domain of } f(g(x)) = \mathbb{R} \]

### Step 3: Find the Range of \( f(g(x)) \)
- The range of \( \sin x \) is \([-1, 1]\).
- Now, we analyze the greatest integer function applied to this range:
- When \( \sin x = 1 \), \( [\sin x] = 1 \).
- When \( 0 \leq \sin x < 1 \), \( [\sin x] = 0 \).
- When \(-1 < \sin x < 0\), \( [\sin x] = -1 \).
- When \( \sin x = -1 \), \( [\sin x] = -1 \).

Thus, the range of \( f(g(x)) = [\sin x] \) is:
\[ \text{Range of } f(g(x)) = \{-1, 0, 1\} \]

### Step 4: Define \( g(f(x)) \) or \( g \circ f(x) \)
To find \( g(f(x)) \):
- Substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g([x]) = \sin([x]) \]
This means we take the sine of the greatest integer less than or equal to \( x \).

### Step 5: Find the Domain of \( g(f(x)) \)
- Since \( f(x) = [x] \) is defined for all real numbers \( x \), and \( g(x) = \sin x \) is also defined for all real numbers, the domain of \( g(f(x)) \) is:
\[ \text{Domain of } g(f(x)) = \mathbb{R} \]

### Step 6: Find the Range of \( g(f(x)) \)
- The output of \( f(x) = [x] \) is an integer. Thus, we need to evaluate \( \sin(n) \) where \( n \) is any integer.
- The range of \( \sin(n) \) for integer values \( n \) is:
\[ \text{Range of } g(f(x)) = \{ \sin(n) \mid n \in \mathbb{Z} \} \]
This means the range consists of the sine values of all integers.

### Summary of Results:
- \( f(g(x)) = [\sin x] \)
- Domain: \( \mathbb{R} \)
- Range: \(\{-1, 0, 1\}\)

- \( g(f(x)) = \sin([x]) \)
- Domain: \( \mathbb{R} \)
- Range: \(\{ \sin(n) \mid n \in \mathbb{Z} \}\)