consider the function `f:R rarr R,f(x)=(x^(2)-6x+4)/(x^(2)+2x+4)`
Range of fX() is

Options

A. `((-oo,-(2)/(3))] cup[2,0)`
B. `[(-1)/(3),5]`
C. `(-oo,2)cup[(7)/(3),oo)`
D. `(20,oo)`

Explanation

To find the range of the function \( f(x) = \frac{x^2 - 6x + 4}{x^2 + 2x + 4} \), we will follow these steps:

### Step 1: Set up the equation
We start by letting \( f(x) = y \):
\[ y = \frac{x^2 - 6x + 4}{x^2 + 2x + 4} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ y(x^2 + 2x + 4) = x^2 - 6x + 4 \]
This simplifies to:
\[ yx^2 + 2yx + 4y = x^2 - 6x + 4 \]

### Step 3: Rearrange the equation
Rearranging the equation leads to:
\[ (y - 1)x^2 + (2y + 6)x + (4y - 4) = 0 \]
This is a quadratic equation in \( x \).

### Step 4: Analyze the discriminant
For \( f(x) \) to take real values, the discriminant of this quadratic must be non-negative:
\[ D = b^2 - 4ac = (2y + 6)^2 - 4(y - 1)(4y - 4) \geq 0 \]

### Step 5: Expand the discriminant
Calculating the discriminant:
\[ D = (2y + 6)^2 - 4(y - 1)(4y - 4) \]
Expanding this gives:
\[ D = 4y^2 + 24y + 36 - 4[(4y^2 - 4y - 4y + 4)] \]
\[ = 4y^2 + 24y + 36 - 4(4y^2 - 8y + 4) \]
\[ = 4y^2 + 24y + 36 - 16y^2 + 32y - 16 \]
\[ = -12y^2 + 56y + 20 \]

### Step 6: Set the discriminant greater than or equal to zero
Now we need to solve the inequality:
\[ -12y^2 + 56y + 20 \geq 0 \]

### Step 7: Factor the quadratic
To find the roots, we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = -12, b = 56, c = 20 \):
\[ D = 56^2 - 4 \cdot (-12) \cdot 20 = 3136 + 960 = 4096 \]
\[ y = \frac{-56 \pm \sqrt{4096}}{2 \cdot -12} = \frac{-56 \pm 64}{-24} \]
Calculating the roots:
1. \( y_1 = \frac{8}{-24} = -\frac{1}{3} \)
2. \( y_2 = \frac{-120}{-24} = 5 \)

### Step 8: Determine the intervals
The quadratic opens downwards (as the coefficient of \( y^2 \) is negative), so the function is non-negative between the roots:
\[ -\frac{1}{3} \leq y \leq 5 \]

### Step 9: Conclusion
Thus, the range of the function \( f(x) \) is:
\[ \boxed{[-\frac{1}{3}, 5]} \]