Find the range of rational expression `y =(x^(2) -x+4)/(x^(2) +x+4)` is x is real.

Answer

To find the range of the rational expression \( y = \frac{x^2 - x + 4}{x^2 + x + 4} \) for real values of \( x \), we will follow these steps:

### Step 1: Rearranging the Expression
We start by rearranging the equation:
\[ y(x^2 + x + 4) = x^2 - x + 4 \]
This simplifies to:
\[ yx^2 + yx + 4y = x^2 - x + 4 \]
Rearranging gives us:
\[ (y - 1)x^2 + (y + 1)x + (4y - 4) = 0 \]

### Step 2: Identifying the Quadratic Form
The expression is now a quadratic equation in \( x \):
\[ (y - 1)x^2 + (y + 1)x + (4y - 4) = 0 \]
For \( x \) to have real solutions, the discriminant of this quadratic must be non-negative.

### Step 3: Finding the Discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = y - 1 \), \( b = y + 1 \), and \( c = 4y - 4 \). Thus,
\[ D = (y + 1)^2 - 4(y - 1)(4y - 4) \]

### Step 4: Simplifying the Discriminant
Calculating \( D \):
\[ D = (y + 1)^2 - 4(y - 1)(4y - 4) \]
Expanding the second term:
\[ D = (y + 1)^2 - 4[(y - 1)(4y - 4)] \]
\[ = (y + 1)^2 - 4(4y^2 - 4y - 4y + 4) \]
\[ = (y + 1)^2 - 16y + 16 \]
Now, simplifying \( D \):
\[ D = y^2 + 2y + 1 - 16y + 16 = y^2 - 14y + 17 \]

### Step 5: Setting the Discriminant Non-Negative
To ensure real solutions exist, we set the discriminant \( D \) greater than or equal to zero:
\[ y^2 - 14y + 17 \geq 0 \]

### Step 6: Finding Roots of the Quadratic
We can find the roots of the quadratic equation \( y^2 - 14y + 17 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 17}}{2 \cdot 1} \]
\[ = \frac{14 \pm \sqrt{196 - 68}}{2} = \frac{14 \pm \sqrt{128}}{2} = \frac{14 \pm 8\sqrt{2}}{2} = 7 \pm 4\sqrt{2} \]

### Step 7: Analyzing the Quadratic Inequality
The roots are \( y_1 = 7 - 4\sqrt{2} \) and \( y_2 = 7 + 4\sqrt{2} \). The quadratic opens upwards (as the coefficient of \( y^2 \) is positive), so the expression \( y^2 - 14y + 17 \geq 0 \) is satisfied outside the interval \( [7 - 4\sqrt{2}, 7 + 4\sqrt{2}] \).

### Step 8: Writing the Final Range
Thus, the range of \( y \) is:
\[ y \leq 7 - 4\sqrt{2} \quad \text{or} \quad y \geq 7 + 4\sqrt{2} \]