What is the equivalent definition of the function. given by f(x) =`{{:(2x,x le0),(0,x lt 0):}`

Options

A. f(x)=|x|
B. f(x)=2x
C. f(x)=|x|+x
D. f(x)=2|x|

Answer

To find the equivalent definition of the function given by \( f(x) = \begin{cases} 2x & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} \), we can express it in a different form.

### Step-by-Step Solution:

1. **Identify the function definition**: The function is defined piecewise. For \( x \geq 0 \), \( f(x) = 2x \) and for \( x < 0 \), \( f(x) = 0 \).

2. **Rewrite the function**: We can express the function using the absolute value function. The absolute value function \( |x| \) is defined as:
\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

3. **Express \( f(x) \) using absolute value**:
- When \( x \geq 0 \):
\[ f(x) = 2x = x + x = x + |x| \]
- When \( x < 0 \):
\[ f(x) = 0 = x + (-x) = x + |x| \]

4. **Combine the results**: We can combine both cases into a single expression:
\[ f(x) = x + |x| \]

5. **Final equivalent definition**: Therefore, the equivalent definition of the function is:
\[ f(x) = x + |x| \]