Let `|__x__|` be defined to be the ''floor'' of x, where `|__x__|` is the greatest integer that is less than or equal to x, and let `|~x~|` be the ''ceiling'' of x, where `|~x~|` is the least integer that is greater than or equal to x. If `f(x) |~x~| + |__x__|` and x is not an integer, than f(x) is also equal to

Options

A. `2|~x~|-2`
B. `2|~x~|`
C. `2|__x__|`
D. `2|__x__|+1`

Explanation

To solve the problem, we need to analyze the functions defined in the question: the floor function \( |__x__| \) and the ceiling function \( |~x~| \).

### Step-by-Step Solution:

1. **Understanding the Floor and Ceiling Functions**:
- The floor function \( |__x__| \) gives the greatest integer less than or equal to \( x \).
- The ceiling function \( |~x~| \) gives the least integer greater than or equal to \( x \).

2. **Considering Non-Integer Values**:
- Since \( x \) is not an integer, we can express \( x \) as \( n + d \), where \( n \) is an integer and \( 0 < d < 1 \) (i.e., \( d \) is the fractional part of \( x \)).
- For example, if \( x = 2.3 \), then \( n = 2 \) and \( d = 0.3 \).

3. **Calculating the Floor and Ceiling Functions**:
- For \( x = n + d \):
- The floor function \( |__x__| = n \) (the greatest integer less than or equal to \( x \)).
- The ceiling function \( |~x~| = n + 1 \) (the least integer greater than or equal to \( x \)).

4. **Finding the Function \( f(x) \)**:
- The function \( f(x) \) is defined as:
\[ f(x) = |~x~| + |__x__| \]
- Substituting the values we found:
\[ f(x) = (n + 1) + n = 2n + 1 \]

5. **Expressing \( f(x) \) in terms of the Floor Function**:
- Since \( n = |__x__| \), we can rewrite \( f(x) \):
\[ f(x) = 2|__x__| + 1 \]

### Conclusion:
Thus, if \( x \) is not an integer, \( f(x) \) can be expressed as:
\[ f(x) = 2|__x__| + 1 \]