Find the range of the following functions: (where {.} and [.] represent fractional part and greatest integer part functions respectively)
`f(x)=(x+2)/(x^(2)-8x-4)`

Explanation

To find the range of the function \( f(x) = \frac{x + 2}{x^2 - 8x - 4} \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
We start by letting \( f(x) = y \):
\[ y = \frac{x + 2}{x^2 - 8x - 4} \]

### Step 2: Rearrange the equation
To express \( x \) in terms of \( y \), we multiply both sides by the denominator:
\[ y(x^2 - 8x - 4) = x + 2 \]
This simplifies to:
\[ yx^2 - 8yx - x - 4y = 2 \]
Rearranging gives:
\[ yx^2 - (8y + 1)x - (4y + 2) = 0 \]

### Step 3: Identify coefficients for the quadratic formula
The quadratic equation in standard form is:
\[ ax^2 + bx + c = 0 \]
where:
- \( a = y \)
- \( b = -(8y + 1) \)
- \( c = -(4y + 2) \)

### Step 4: Apply the quadratic formula
To find \( x \), we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{8y + 1 \pm \sqrt{(8y + 1)^2 - 4y(-4y - 2)}}{2y} \]

### Step 5: Ensure the discriminant is non-negative
For \( x \) to be real, the discriminant must be greater than or equal to zero:
\[ (8y + 1)^2 + 16y^2 + 8y \geq 0 \]
This simplifies to:
\[ 64y^2 + 16y + 1 + 16y^2 + 8y \geq 0 \]
Combining like terms:
\[ 80y^2 + 24y + 1 \geq 0 \]

### Step 6: Factor the quadratic
Now, we can factor or use the quadratic formula to find the roots of:
\[ 80y^2 + 24y + 1 = 0 \]
Using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 80 \cdot 1}}{2 \cdot 80} \]
Calculating the discriminant:
\[ 24^2 - 320 = 576 - 320 = 256 \]
Thus:
\[ y = \frac{-24 \pm 16}{160} \]
Calculating the roots:
1. \( y_1 = \frac{-8}{160} = -\frac{1}{20} \)
2. \( y_2 = \frac{-40}{160} = -\frac{1}{4} \)

### Step 7: Determine the intervals for the range
The quadratic \( 80y^2 + 24y + 1 \) opens upwards (since the coefficient of \( y^2 \) is positive). The expression is non-negative outside the roots:
- \( y \leq -\frac{1}{4} \) or \( y \geq -\frac{1}{20} \)

### Final Step: Write the range
Thus, the range of the function \( f(x) \) is:
\[ (-\infty, -\frac{1}{4}] \cup [-\frac{1}{20}, \infty) \]