Redefine the function `f(x)= |1+x| +|1-x|, -2 le x le 2`.

Answer

To redefine the function \( f(x) = |1+x| + |1-x| \) for the interval \(-2 \leq x \leq 2\), we will analyze the function by considering the critical points where the expressions inside the absolute values change sign. The critical points for this function are \( x = -1 \) and \( x = 1 \).

### Step 1: Identify the intervals
The critical points divide the interval \([-2, 2]\) into three intervals:
1. \( -2 \leq x < -1 \)
2. \( -1 \leq x < 1 \)
3. \( 1 \leq x \leq 2 \)

### Step 2: Analyze each interval

**Interval 1: \( -2 \leq x < -1 \)**
- In this interval, both \( 1+x \) and \( 1-x \) are negative.
- Therefore, we have:
\[ f(x) = |1+x| + |1-x| = -(1+x) - (1-x) = -1 - x - 1 + x = -2 \]

**Interval 2: \( -1 \leq x < 1 \)**
- In this interval, \( 1+x \) is non-negative and \( 1-x \) is non-negative.
- Therefore, we have:
\[ f(x) = |1+x| + |1-x| = (1+x) + (1-x) = 1 + x + 1 - x = 2 \]

**Interval 3: \( 1 \leq x \leq 2 \)**
- In this interval, \( 1+x \) is positive and \( 1-x \) is negative.
- Therefore, we have:
\[ f(x) = |1+x| + |1-x| = (1+x) - (1-x) = 1 + x - 1 + x = 2x \]

### Step 3: Combine the results
Now we can combine the results from each interval to redefine the function \( f(x) \):
\[ f(x) = \begin{cases} -2 & \text{if } -2 \leq x < -1 \\ 2 & \text{if } -1 \leq x < 1 \\ 2x & \text{if } 1 \leq x \leq 2 \end{cases} \]

### Final Answer
Thus, the redefined function is:
\[ f(x) = \begin{cases} -2 & \text{if } -2 \leq x < -1 \\ 2 & \text{if } -1 \leq x < 1 \\ 2x & \text{if } 1 \leq x \leq 2 \end{cases} \]