Let `f(x)={(-2,-3 lex le0),(x-2,0 lt x le3):}`, where `g(x) = min {f(|x|)+|f(x)|, f(|x|)-|f(x)|}`. Find the area bounded by the curve `g(x)` and the X-axis between the ordinates at `x=3` and `x=-3`.

Answer

To solve the problem, we need to find the area bounded by the curve \( g(x) \) and the X-axis between the ordinates at \( x = 3 \) and \( x = -3 \).

### Step-by-Step Solution:

1. **Define the function \( f(x) \)**:
\[ f(x) = \begin{cases} -2 & \text{for } -3 \leq x \leq 0 \\ x - 2 & \text{for } 0 < x \leq 3 \end{cases} \]

2. **Determine \( f(|x|) \)**:
- For \( x < 0 \) (i.e., \( -3 \leq x < 0 \)):
\[ f(|x|) = f(-x) = f(-(-x)) = f(x) = -2 \]
- For \( x = 0 \):
\[ f(|0|) = f(0) = -2 \]
- For \( x > 0 \) (i.e., \( 0 < x \leq 3 \)):
\[ f(|x|) = f(x) = x - 2 \]

3. **Determine \( |f(x)| \)**:
- For \( -3 \leq x < 0 \):
\[ |f(x)| = |-2| = 2 \]
- For \( 0 < x \leq 3 \):
\[ |f(x)| = |x - 2| = \begin{cases} 2 - x & \text{for } 0 < x < 2 \\ x - 2 & \text{for } 2 \leq x \leq 3 \end{cases} \]

4. **Define \( g(x) \)**:
\[ g(x) = \min \{ f(|x|) + |f(x)|, f(|x|) - |f(x)| \} \]

5. **Calculate \( g(x) \) in different intervals**:
- For \( -3 \leq x < 0 \):
\[ g(x) = \min \{ -2 + 2, -2 - 2 \} = \min \{ 0, -4 \} = -4 \]
- For \( 0 < x < 2 \):
\[ g(x) = \min \{ x - 2 + (2 - x), x - 2 - (2 - x) \} = \min \{ 0, 2x - 4 \} \]
- Here, \( 2x - 4 \) is negative for \( x < 2 \), so \( g(x) = 0 \).
- For \( 2 \leq x \leq 3 \):
\[ g(x) = \min \{ x - 2 + (x - 2), x - 2 - (x - 2) \} = \min \{ 2x - 4, 0 \} \]
- Here, \( 2x - 4 \) is non-negative for \( x \geq 2 \), so \( g(x) = 0 \).

6. **Sketch the graph of \( g(x) \)**:
- From \( -3 \) to \( 0 \), \( g(x) = -4 \).
- From \( 0 \) to \( 2 \), \( g(x) = 0 \).
- From \( 2 \) to \( 3 \), \( g(x) = 0 \).

7. **Calculate the area bounded by \( g(x) \) and the X-axis**:
- The area from \( x = -3 \) to \( x = 0 \) is a rectangle:
\[ \text{Area} = \text{base} \times \text{height} = (0 - (-3)) \times 4 = 3 \times 4 = 12 \]
- The area from \( x = 0 \) to \( x = 3 \) is zero since \( g(x) = 0 \).

8. **Total Area**:
\[ \text{Total Area} = 12 + 0 = 12 \]

### Final Answer:
The area bounded by the curve \( g(x) \) and the X-axis between the ordinates at \( x = 3 \) and \( x = -3 \) is \( 12 \).