If `f(x)={(2+2x,-1 le x lt 0),(1-x/(3),0 le x le 3))`
`g(x)={(-x,-3 le x le 0),(x, 0 lt x le 1))`, then range of `(fog)(x)` is

Options

A. [0,3)
B. [0,1)
C. [0,1]
D. (0,1]

Answer

To find the range of the composite function \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \) based on the definitions of the functions \( f(x) \) and \( g(x) \).

### Step 1: Define the functions
The functions are given as:
- \( f(x) = \begin{cases} 2 + 2x & \text{if } -1 \leq x < 0 \\ 1 - \frac{x}{3} & \text{if } 0 \leq x \leq 3 \end{cases} \)

- \( g(x) = \begin{cases} -x & \text{if } -3 \leq x \leq 0 \\ x & \text{if } 0 < x \leq 1 \end{cases} \)

### Step 2: Determine the range of \( g(x) \)
1. For \( -3 \leq x \leq 0 \):
- \( g(x) = -x \) ranges from \( g(-3) = 3 \) to \( g(0) = 0 \).
- So, \( g(x) \) in this interval gives us the range \( [0, 3] \).

2. For \( 0 < x \leq 1 \):
- \( g(x) = x \) ranges from \( g(0) = 0 \) to \( g(1) = 1 \).
- So, \( g(x) \) in this interval gives us the range \( (0, 1] \).

Combining both intervals, the overall range of \( g(x) \) is \( [0, 3] \).

### Step 3: Evaluate \( f(g(x)) \)
Now we need to evaluate \( f(g(x)) \) for the range \( [0, 3] \).

1. **For \( g(x) \) in \( [0, 3] \)**:
- When \( 0 \leq g(x) < 0 \):
- This part does not apply since \( g(x) \) does not take negative values in the range we found.
- When \( 0 \leq g(x) \leq 3 \):
- We will use the second part of \( f(x) \):
\[ f(g(x)) = 1 - \frac{g(x)}{3} \]

### Step 4: Determine the range of \( f(g(x)) \)
1. **Evaluate at the endpoints**:
- For \( g(x) = 0 \):
\[ f(0) = 1 - \frac{0}{3} = 1 \]
- For \( g(x) = 3 \):
\[ f(3) = 1 - \frac{3}{3} = 0 \]

2. **Range of \( f(g(x)) \)**:
- As \( g(x) \) varies from \( 0 \) to \( 3 \), \( f(g(x)) \) decreases linearly from \( 1 \) to \( 0 \).
- Thus, the range of \( f(g(x)) \) is \( [0, 1] \).

### Conclusion
The range of the composite function \( (f \circ g)(x) \) is \( [0, 1] \).