The equivalent definition of `f(x)=||x|-1|`, is

Options

A. `f(x)={{:( -x-1, x le -1),(x+1, -1 lt x le 0),( 1-x, 0 le x le 1), (x-1,x le 1):}`
B. `f(x)={{:(x-1,x le -1),(x+1,-1 lt x le 0),(x-1, 0 le x le 1),(x+1, x ge 1):}`
C. `f(x)={{:( x+1, x ge 0), (x+1, x le0):}`
D. none of these

Answer

To find the equivalent definition of the function \( f(x) = ||x| - 1| \), we will analyze the function step by step.

### Step 1: Understanding the Inner Modulus
The function \( f(x) = ||x| - 1| \) consists of two modulus operations. The inner modulus \( |x| - 1 \) needs to be evaluated first.

**Hint:** Recall that the modulus function \( |a| \) is defined as:
- \( a \) if \( a \geq 0 \)
- \( -a \) if \( a < 0 \)

### Step 2: Analyzing \( |x| \)
The expression \( |x| \) is non-negative for all \( x \). Therefore, we can break it down into cases based on the value of \( x \).

1. **Case 1:** \( x \geq 0 \)
- Here, \( |x| = x \)
- Thus, \( |x| - 1 = x - 1 \)

2. **Case 2:** \( x < 0 \)
- Here, \( |x| = -x \)
- Thus, \( |x| - 1 = -x - 1 \)

### Step 3: Evaluating \( ||x| - 1| \)
Now we will evaluate \( ||x| - 1| \) for both cases.

1. **For \( x \geq 0 \):**
- We need to evaluate \( |x - 1| \).
- This can be broken into two sub-cases:
- If \( x \geq 1 \): \( |x - 1| = x - 1 \)
- If \( 0 \leq x < 1 \): \( |x - 1| = 1 - x \)

2. **For \( x < 0 \):**
- We need to evaluate \( |-x - 1| \).
- This can also be broken into two sub-cases:
- If \( -x - 1 \geq 0 \) (which simplifies to \( x \leq -1 \)): \( |-x - 1| = -(-x - 1) = x + 1 \)
- If \( -x - 1 < 0 \) (which simplifies to \( -1 < x < 0 \)): \( |-x - 1| = -(-x - 1) = -x - 1 \)

### Step 4: Combining All Cases
Now we can summarize the function \( f(x) \) based on the cases we have analyzed:

1. **If \( x \leq -1 \):**
- \( f(x) = x + 1 \)

2. **If \( -1 < x < 0 \):**
- \( f(x) = -x - 1 \)

3. **If \( 0 \leq x < 1 \):**
- \( f(x) = 1 - x \)

4. **If \( x \geq 1 \):**
- \( f(x) = x - 1 \)

### Final Result
Thus, the equivalent definition of the function \( f(x) = ||x| - 1| \) can be expressed as:

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