Let `f(x)=f_1(x)-2f_2 (x)`, where ,where `f_1(x)={((min{x^2,|x|},|x|le 1),(max{x^2,|x|},|x| le 1))` and `f_2(x)={((min{x^2,|x|},|x| lt 1),({x^2,|x|},|x| le 1))` and let `g(x)={ ((min{f(t):-3letlex,-3 le x le 0}),(max{f(t):0 le t le x,0 le x le 3}))` for `-3 le x le -1` the range of `g(x)` is

Options

A. `[-1, 3]`
B. `[-1,-15]`
C. `[-1, 9]`
D. None of these

Explanation

To solve the problem step by step, we will analyze the functions \( f_1(x) \), \( f_2(x) \), and \( g(x) \) as given in the question.

### Step 1: Define \( f_1(x) \) and \( f_2(x) \)

1. **For \( f_1(x) \)**:
- When \( |x| < 1 \): \( f_1(x) = \min(x^2, |x|) \)
- When \( |x| \leq 1 \): \( f_1(x) = \max(x^2, |x|) \)

2. **For \( f_2(x) \)**:
- When \( |x| < 1 \): \( f_2(x) = \min(x^2, |x|) \)
- When \( |x| \leq 1 \): \( f_2(x) = x^2 \)

### Step 2: Analyze \( f_1(x) \)

- For \( |x| < 1 \):
- \( x^2 \) is always less than \( |x| \) for \( |x| < 1 \), hence \( f_1(x) = x^2 \).

- For \( |x| = 1 \):
- \( f_1(1) = \max(1^2, 1) = 1 \)
- \( f_1(-1) = \max(1^2, 1) = 1 \)

### Step 3: Analyze \( f_2(x) \)

- For \( |x| < 1 \):
- Similar to \( f_1(x) \), \( f_2(x) = x^2 \).

- For \( |x| = 1 \):
- \( f_2(1) = 1^2 = 1 \)
- \( f_2(-1) = 1^2 = 1 \)

### Step 4: Define \( f(x) \)

Now we can define \( f(x) \):
- For \( |x| < 1 \):
\[ f(x) = f_1(x) - 2f_2(x) = x^2 - 2x^2 = -x^2 \]
- For \( |x| = 1 \):
\[ f(1) = 1 - 2(1) = -1, \quad f(-1) = 1 - 2(1) = -1 \]

### Step 5: Analyze \( g(x) \)

The function \( g(x) \) is defined as:
\[ g(x) = \min\{f(t) : -3 \leq t \leq x\} \text{ for } -3 \leq x \leq -1 \]

### Step 6: Find the range of \( g(x) \)

1. **Evaluate \( f(t) \) for \( t \in [-3, -1] \)**:
- For \( t < -1 \), \( f(t) = -t^2 \).
- At \( t = -1 \), \( f(-1) = -1 \).
- At \( t = -3 \), \( f(-3) = -(-3)^2 = -9 \).

2. **Determine the minimum value of \( f(t) \)**:
- As \( t \) moves from \(-3\) to \(-1\), \( f(t) \) decreases from \(-9\) to \(-1\).
- Hence, the minimum value of \( f(t) \) in this interval is \(-9\) and the maximum is \(-1\).

### Step 7: Conclusion

Thus, the range of \( g(x) \) for \( -3 \leq x \leq -1 \) is:
\[ [-9, -1] \]