Let f(x) be defined on [-2,2] and is given by
f(x) = `{{:(x+1, - 2le x le 0 ),(x - 1, 0 le x le 2):}`, then f (|x|) is defined as

Options

A. f (|x|) = `{{:(-1, - 2le x le 0 ),(x - 1, 0 le x le 2):}`
B. f(|x|) = x -1 `forall x in `R
C. ` f (|x|) = {{:(-x -1, - 2le x le 0 ),(x - 1, 0 le x le 2):}`
D. None of these

Answer

To solve the problem, we need to determine the function \( f(|x|) \) based on the given piecewise function \( f(x) \).

The function \( f(x) \) is defined as follows:

\[ f(x) = \begin{cases} x + 1 & \text{if } -2 \leq x \leq 0 \\ x - 1 & \text{if } 0 < x \leq 2 \end{cases} \]

### Step 1: Determine the range of \( |x| \)

The absolute value function \( |x| \) takes non-negative values. Therefore, we need to consider the intervals for \( |x| \):
- When \( x \) is in the range \( [-2, 0] \), \( |x| \) will range from \( 0 \) to \( 2 \).
- When \( x \) is in the range \( [0, 2] \), \( |x| \) will also range from \( 0 \) to \( 2 \).

Thus, \( |x| \) will take values in the interval \( [0, 2] \).

### Step 2: Evaluate \( f(|x|) \)

Now we can evaluate \( f(|x|) \) based on the definition of \( f(x) \):

1. **For \( |x| \) in the interval \( [0, 2] \)**:
- If \( |x| = 0 \), then we use the first case of \( f(x) \):
\[ f(0) = 0 + 1 = 1 \]
- If \( |x| \) is in the interval \( (0, 2] \), we use the second case of \( f(x) \):
\[ f(|x|) = |x| - 1 \]

### Step 3: Combine the results

Thus, we can write the function \( f(|x|) \) as:

\[ f(|x|) = \begin{cases} 1 & \text{if } |x| = 0 \\ |x| - 1 & \text{if } 0 < |x| \leq 2 \end{cases} \]

### Final Function Representation

To summarize, the function \( f(|x|) \) can be expressed as:

\[ f(|x|) = \begin{cases} 1 & \text{if } |x| = 0 \\ |x| - 1 & \text{if } 0 < |x| \leq 2 \end{cases} \]