Find domain and range of following function: `y=(x^(2)-2x+5)/(x^(2)+2x+5)`

Explanation

To find the domain and range of the function \( y = \frac{x^2 - 2x + 5}{x^2 + 2x + 5} \), we will follow these steps:

### Step 1: Determine the Domain
The domain of a rational function is all real numbers except where the denominator is zero. Therefore, we need to find when the denominator \( x^2 + 2x + 5 \) is equal to zero.

1. Set the denominator equal to zero:
\[ x^2 + 2x + 5 = 0 \]

2. Calculate the discriminant \( D \):
\[ D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16 \]
Since the discriminant is negative, the quadratic equation has no real roots.

3. Therefore, the denominator is never zero for any real number \( x \).

Thus, the domain of the function is:
\[ \text{Domain} = \mathbb{R} \quad \text{(all real numbers)} \]

### Step 2: Determine the Range
To find the range, we will manipulate the equation and analyze it.

1. Cross-multiply to eliminate the fraction:
\[ y(x^2 + 2x + 5) = x^2 - 2x + 5 \]

2. Rearranging gives:
\[ yx^2 + 2yx + 5y - x^2 + 2x - 5 = 0 \]
\[ (y - 1)x^2 + (2y + 2)x + (5y - 5) = 0 \]

3. For \( x \) to have real solutions, the discriminant of this quadratic must be non-negative:
\[ D' = (2y + 2)^2 - 4(y - 1)(5y - 5) \geq 0 \]

4. Expanding the discriminant:
\[ D' = 4(y + 1)^2 - 4(y - 1)(5y - 5) \]
\[ = 4(y + 1)^2 - 4[(y - 1)(5y - 5)] \]
\[ = 4(y + 1)^2 - 4(5y^2 - 10y - 5 + 5) \]
\[ = 4(y + 1)^2 - 20y^2 + 40y - 20 \]

5. Simplifying:
\[ = 4y^2 + 8y + 4 - 20y^2 + 40y - 20 \]
\[ = -16y^2 + 48y - 16 \]

6. Setting this greater than or equal to zero:
\[ -16(y^2 - 3y + 1) \geq 0 \]
\[ y^2 - 3y + 1 \leq 0 \]

7. Finding the roots of the quadratic:
\[ y = \frac{3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{3 \pm \sqrt{5}}{2} \]

8. The roots are \( y_1 = \frac{3 - \sqrt{5}}{2} \) and \( y_2 = \frac{3 + \sqrt{5}}{2} \).

9. The quadratic opens upwards (since the coefficient of \( y^2 \) is positive), thus the range of \( y \) is:
\[ \text{Range} = \left[ \frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2} \right] \]

### Final Answer:
- **Domain**: \( \mathbb{R} \) (all real numbers)
- **Range**: \( \left[ \frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2} \right] \)