`f(x)={(4, xlt-1) ,(-4x,-1lt=xlt=0):}` If `f(x)` is an even function in R then the definition of `f(x)` in `(0,oo)` is: (A) `f(x)={(4x, 0ltxle1),(4, xgt1):}` (B) `f(x)={(4x, 0ltxle1),(-4, xgt1):}` (C) `f(x)={(4, 0ltxle1),(4x, xgt1):}` (D) `f(x)={(4, xlt-1),(-4x, -1lexle0):}`

Options

A. `f(x)={:(4x, 0 lt x le 1),(4, x gt 1):}`
B. `f(x)={{:(4x, 0 lt x le1),(-4, x gt 1):}`
C. `f(x)={{:(4, 0 lt x le 1),(4x,x gt 1):}`
D. none of these

Explanation

To determine the definition of \( f(x) \) in the interval \( (0, \infty) \) given that \( f(x) \) is an even function, we start by analyzing the provided piecewise function:

1. **Given Function**:
\[ f(x) = \begin{cases} 4 & \text{if } x < -1 \\ -4x & \text{if } -1 \leq x < 0 \end{cases} \]

2. **Understanding Even Functions**:
An even function satisfies the property \( f(x) = f(-x) \) for all \( x \). This means that the function's values for positive inputs must mirror the values for their corresponding negative inputs.

3. **Finding \( f(x) \) for \( x \in (0, \infty) \)**:
- For \( x < 0 \):
- When \( -1 \leq x < 0 \), \( f(x) = -4x \).
- Therefore, \( f(-x) = -4(-x) = 4x \) for \( 0 < x \leq 1 \).
- For \( x < -1 \):
- Here, \( f(x) = 4 \).
- Thus, \( f(-x) = 4 \) for \( x > 1 \).

4. **Constructing the Function in \( (0, \infty) \)**:
- From the above analysis, we can conclude:
- For \( 0 < x \leq 1 \), \( f(x) = 4x \).
- For \( x > 1 \), \( f(x) = 4 \).

5. **Final Piecewise Definition**:
Therefore, the definition of \( f(x) \) in the interval \( (0, \infty) \) is:
\[ f(x) = \begin{cases} 4x & \text{if } 0 < x \leq 1 \\ 4 & \text{if } x > 1 \end{cases} \]

6. **Selecting the Correct Option**:
Among the given options, this matches with:
(A) \( f(x) = \{(4x, 0 < x \leq 1), (4, x > 1)\} \).