Let f (x) be defined as
`f (x) ={{:(|x|, 0 le x lt1),(|x-1|+|x-2|, 1 le x lt2),(|x-3|, 2 le x lt 3):}`
The range of function `g (x)= sin (7 (f (x))` is :

Options

A. `[0,1]`
B. `[-1,0]`
C. `[-(1)/(2), (1)/(2)]`
D. `[-1,1]`

Answer

To find the range of the function \( g(x) = \sin(7f(x)) \), we first need to determine the range of the function \( f(x) \) defined in three parts. Let's break down the solution step by step.

### Step 1: Analyze the first part of \( f(x) \)
For \( 0 \leq x < 1 \):
\[ f(x) = |x| = x \]
The range of \( f(x) \) in this interval is:
\[ [0, 1) \]

### Step 2: Analyze the second part of \( f(x) \)
For \( 1 \leq x < 2 \):
\[ f(x) = |x - 1| + |x - 2| \]
We can break this down further:
- For \( 1 < x < 2 \):
\[ f(x) = (x - 1) + (2 - x) = 1 \]
- For \( x = 1 \):
\[ f(1) = |1 - 1| + |1 - 2| = 0 + 1 = 1 \]

Thus, in the interval \( 1 \leq x < 2 \), \( f(x) \) takes the value \( 1 \). Therefore, the range from this part is:
\[ \{1\} \]

### Step 3: Analyze the third part of \( f(x) \)
For \( 2 \leq x < 3 \):
\[ f(x) = |x - 3| \]
In this interval:
- For \( 2 \leq x < 3 \):
\[ f(x) = 3 - x \]
The range of \( f(x) \) in this interval is:
\[ [0, 1) \]

### Step 4: Combine the ranges
Now, we combine the ranges from all three parts:
1. From \( 0 \leq x < 1 \): \( [0, 1) \)
2. From \( 1 \leq x < 2 \): \( \{1\} \)
3. From \( 2 \leq x < 3 \): \( [0, 1) \)

Thus, the overall range of \( f(x) \) is:
\[ [0, 1) \]

### Step 5: Find the range of \( g(x) = \sin(7f(x)) \)
Since \( f(x) \) takes values in the interval \( [0, 1) \), we can find the range of \( g(x) \):
- The minimum value of \( 7f(x) \) is \( 0 \) (when \( f(x) = 0 \)).
- The maximum value of \( 7f(x) \) is \( 7 \) (as \( f(x) \) approaches \( 1 \)).

Now we evaluate \( g(x) = \sin(7f(x)) \):
- When \( 7f(x) = 0 \), \( g(x) = \sin(0) = 0 \).
- When \( 7f(x) \) approaches \( 7 \), we need to find \( \sin(7) \).

### Step 6: Determine the range of \( g(x) \)
The sine function oscillates between \(-1\) and \(1\). Since \( 7 \) is less than \( 2\pi \) (approximately \( 6.28 \)), we can determine that:
\[ \sin(7) \text{ is a value between } -1 \text{ and } 1. \]

Thus, the range of \( g(x) \) is:
\[ [-1, 1] \]

### Final Answer
The range of the function \( g(x) = \sin(7f(x)) \) is:
\[ [-1, 1] \]