Let `f:R-{2,6} rarr R` be real valued function defined as `f(x)=(x^(2)+2x+1)/(x^(2)-8x+12).` Then range of `f` is

Options

A. `(-oo,-(21)/(4)|uu[(21)/(4),oo)`
B. `(-oo,-(21)/(4)]uu[0,oo)`
C. `(-oo,-(21)/(4))uu(0,oo)`
D. `(-oo,-(21)/(4)]uu(1,oo)`

Answer

To find the range of the function \( f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12} \), we will follow these steps:

### Step 1: Rewrite the function
We can rewrite the function as:
\[ f(x) = \frac{(x + 1)^2}{(x - 2)(x - 6)} \]

### Step 2: Set \( f(x) = y \)
We will set \( f(x) = y \) and cross-multiply:
\[ y(x^2 - 8x + 12) = x^2 + 2x + 1 \]
This leads to:
\[ yx^2 - 8yx + 12y = x^2 + 2x + 1 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ (y - 1)x^2 + (-8y - 2)x + (12y - 1) = 0 \]

### Step 4: Determine the discriminant
For \( f(x) \) to have real values, the discriminant of this quadratic equation must be non-negative:
\[ D = (-8y - 2)^2 - 4(y - 1)(12y - 1) \geq 0 \]

### Step 5: Calculate the discriminant
Calculating the discriminant:
\[ D = (64y^2 + 32y + 4) - 4[(y - 1)(12y - 1)] \]
Expanding the second term:
\[ D = 64y^2 + 32y + 4 - 4(12y^2 - 13y + 1) \]
This simplifies to:
\[ D = 64y^2 + 32y + 4 - 48y^2 + 52y - 4 \]
Combining like terms:
\[ D = 16y^2 + 84y \]

### Step 6: Factor the discriminant
Factoring gives:
\[ D = 4y(4y + 21) \geq 0 \]

### Step 7: Solve the inequality
The solutions to this inequality are:
1. \( y \leq 0 \) (from \( 4y \geq 0 \))
2. \( y \geq -\frac{21}{4} \) (from \( 4y + 21 \geq 0 \))

### Step 8: Exclude \( y = 1 \)
Since \( f(x) \) cannot equal 1 (as shown in the video), we exclude this value from our range.

### Step 9: Final range
Thus, the range of \( f(x) \) is:
\[ (-\infty, -\frac{21}{4}] \cup (0, \infty) \quad \text{except } y = 1 \]

### Final Answer
The range of \( f \) is:
\[ (-\infty, -\frac{21}{4}] \cup (0, 1) \cup (1, \infty) \]