find the range of the function `f(x)=1/(2{-x})-{x}` occurs at x equals where {.} represents the fractional part functional

Explanation

To find the range of the function \( f(x) = \frac{1}{2\{ -x \}} - x \), where \( \{ . \} \) represents the fractional part function, we can follow these steps:

### Step 1: Understand the Fractional Part Function
The fractional part function \( \{ x \} \) is defined as \( x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \). Thus, \( \{ -x \} = 1 - \{ x \} \) when \( x \) is not an integer.

### Step 2: Rewrite the Function
Using the property of the fractional part, we can rewrite the function:
\[ f(x) = \frac{1}{2(1 - \{ x \})} - x \]
This means we can express \( f(x) \) as:
\[ f(x) = \frac{1}{2(1 - \{ x \})} - x \]

### Step 3: Substitute \( y = \{ x \} \)
Since \( \{ x \} \) can take values in the interval \( [0, 1) \), we can let \( y = \{ x \} \). Therefore, we can rewrite the function in terms of \( y \):
\[ f(y) = \frac{1}{2(1 - y)} - (n + y) \quad \text{(where \( n = \lfloor x \rfloor \))} \]
This simplifies to:
\[ f(y) = \frac{1}{2(1 - y)} - n - y \]

### Step 4: Analyze the Function
To analyze the function, we need to consider the behavior of \( f(y) \) as \( y \) varies from \( 0 \) to \( 1 \):
1. As \( y \to 0 \):
\[ f(0) = \frac{1}{2(1 - 0)} - n = \frac{1}{2} - n \]
2. As \( y \to 1 \):
\[ f(1) \text{ is undefined since } \{ x \} \text{ cannot be } 1. \]

### Step 5: Find Critical Points
To find the minimum value, we differentiate \( f(y) \) with respect to \( y \) and set the derivative to zero:
\[ f'(y) = \frac{d}{dy}\left(\frac{1}{2(1 - y)} - n - y\right) \]
Calculating the derivative:
\[ f'(y) = \frac{1}{2} \cdot \frac{1}{(1 - y)^2} - 1 \]
Setting \( f'(y) = 0 \):
\[ \frac{1}{2(1 - y)^2} = 1 \implies (1 - y)^2 = \frac{1}{2} \implies 1 - y = \frac{1}{\sqrt{2}} \implies y = 1 - \frac{1}{\sqrt{2}} \]

### Step 6: Evaluate \( f(y) \) at Critical Points
Substituting \( y = 1 - \frac{1}{\sqrt{2}} \) back into \( f(y) \):
\[ f\left(1 - \frac{1}{\sqrt{2}}\right) = \frac{1}{2\left(\frac{1}{\sqrt{2}}\right)} - n - \left(1 - \frac{1}{\sqrt{2}}\right) \]
This simplifies to:
\[ f\left(1 - \frac{1}{\sqrt{2}}\right) = \frac{\sqrt{2}}{2} - n - 1 + \frac{1}{\sqrt{2}} \]

### Step 7: Determine the Range
As \( n \) can take any integer value, the minimum value of \( f(y) \) will approach \( \sqrt{2} - 1 \) as \( n \) increases. Thus, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = \left[\sqrt{2} - 1, \infty\right) \]

### Final Answer
The range of the function \( f(x) = \frac{1}{2\{-x\}} - x \) is:
\[ \boxed{[\sqrt{2} - 1, \infty)} \]