Find the range of `f(x)=(x^(2)+14x+9)/(x^(2)+2x+3)`, where x `in` R.

Answer

To find the range of the function \( f(x) = \frac{x^2 + 14x + 9}{x^2 + 2x + 3} \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
Let \( y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3} \).

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ y(x^2 + 2x + 3) = x^2 + 14x + 9 \]
This simplifies to:
\[ yx^2 + 2yx + 3y = x^2 + 14x + 9 \]

### Step 3: Rearrange into a standard quadratic form
Rearranging the equation, we get:
\[ (y - 1)x^2 + (2y - 14)x + (3y - 9) = 0 \]
This is a quadratic equation in \( x \).

### Step 4: Analyze the discriminant
For the quadratic equation to have real solutions, the discriminant must be greater than or equal to zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
where \( a = y - 1 \), \( b = 2y - 14 \), and \( c = 3y - 9 \).

Calculating the discriminant:
\[ D = (2y - 14)^2 - 4(y - 1)(3y - 9) \]

### Step 5: Expand and simplify the discriminant
Expanding \( D \):
\[ D = (2y - 14)^2 - 4[(y - 1)(3y - 9)] \]
\[ D = 4y^2 - 56y + 196 - 4[(3y^2 - 9y - 3y + 9)] \]
\[ D = 4y^2 - 56y + 196 - 4(3y^2 - 12y + 9) \]
\[ D = 4y^2 - 56y + 196 - 12y^2 + 48y - 36 \]
\[ D = -8y^2 - 8y + 160 \]

### Step 6: Set the discriminant greater than or equal to zero
Setting the discriminant \( D \geq 0 \):
\[ -8y^2 - 8y + 160 \geq 0 \]
Dividing by -8 (which reverses the inequality):
\[ y^2 + y - 20 \leq 0 \]

### Step 7: Factor the quadratic
Factoring gives us:
\[ (y + 5)(y - 4) \leq 0 \]

### Step 8: Find the critical points
The critical points are \( y = -5 \) and \( y = 4 \).

### Step 9: Test intervals
Testing intervals around the critical points:
- For \( y < -5 \): Choose \( y = -6 \) → \( (-)(-) > 0 \)
- For \( -5 < y < 4 \): Choose \( y = 0 \) → \( (+)(-) < 0 \)
- For \( y > 4 \): Choose \( y = 5 \) → \( (+)(+) > 0 \)

### Step 10: Conclusion
The solution to the inequality \( (y + 5)(y - 4) \leq 0 \) gives us:
\[ -5 \leq y \leq 4 \]
Thus, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = [-5, 4] \]