Let `f (x), g(x)` be two real valued functions then the function `h(x) =2 max {f(x)-g(x), 0}` is equal to :

Options

A. `f (x) -g(x) -|g(x)-f(x)|`
B. `f(x) +g(x) -|g(x)- f(x)|`
C. `f(x) -g (x)+ |g(x) -f(x)|`
D. `f (x)+g (x)+ |g(x)- f(x)|`

Answer

To solve the problem, we need to analyze the function \( h(x) = 2 \max \{ f(x) - g(x), 0 \} \).

### Step-by-Step Solution:

1. **Understanding the max function**:
The expression \( \max \{ f(x) - g(x), 0 \} \) means we take the larger value between \( f(x) - g(x) \) and \( 0 \). This can be interpreted as:
- If \( f(x) - g(x) \geq 0 \), then \( \max \{ f(x) - g(x), 0 \} = f(x) - g(x) \).
- If \( f(x) - g(x) < 0 \), then \( \max \{ f(x) - g(x), 0 \} = 0 \).

2. **Expressing \( h(x) \)**:
Based on the definition of the max function, we can rewrite \( h(x) \) in two cases:
- Case 1: If \( f(x) - g(x) \geq 0 \):
\[ h(x) = 2 \cdot (f(x) - g(x)) \]
- Case 2: If \( f(x) - g(x) < 0 \):
\[ h(x) = 2 \cdot 0 = 0 \]

3. **Combining the cases**:
We can summarize the function \( h(x) \) as:
\[ h(x) = \begin{cases} 2(f(x) - g(x)) & \text{if } f(x) - g(x) \geq 0 \\ 0 & \text{if } f(x) - g(x) < 0 \end{cases} \]

4. **Final expression**:
We can express \( h(x) \) in a more compact form:
\[ h(x) = 2 \max \{ f(x) - g(x), 0 \} \]
This indicates that \( h(x) \) will be twice the positive difference between \( f(x) \) and \( g(x) \) when \( f(x) \) is greater than or equal to \( g(x) \), and zero otherwise.

### Conclusion:
Thus, the function \( h(x) \) can be expressed as:
\[ h(x) = 2 \max \{ f(x) - g(x), 0 \} \]
This is the final answer.