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The problem asks to find the range of the function:

\[
y = \frac{x^2 - 3x + 4}{x^2 + 3x + 4}.
\]

### Step-by-step Solution:

1. **Identify the Function and Simplify:**  
   We need to analyze the function \( y = \frac{x^2 - 3x + 4}{x^2 + 3x + 4} \). We observe that both the numerator and the denominator are quadratic expressions in terms of \(x\).

2. **Check for Denominator Zero:**  
   The denominator \(x^2 + 3x + 4\) must not be zero. We can find its discriminant to ensure it has no real roots:
   
   \[
   \text{Discriminant} = (3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7.
   \]
   
   Since the discriminant is negative, the denominator never equals zero, so there are no restrictions from the domain.

3. **Analyze the Behavior of the Function:**  
   Let's set \( y = \frac{x^2 - 3x + 4}{x^2 + 3x + 4} \) and try to express \(x\) in terms of \(y\). Rearranging the equation gives:
   
   \[
   y (x^2 + 3x + 4) = x^2 - 3x + 4,
   \]
   
   simplifying to:
   
   \[
   yx^2 + 3yx + 4y = x^2 - 3x + 4.
   \]
   
   Rearranging terms:
   
   \[
   (y - 1)x^2 + (3y + 3)x + (4y - 4) = 0.
   \]
   
4. **Find the Conditions for Real Roots:**  
   For \(x\) to have real values, the discriminant of the resulting quadratic equation must be non-negative:
   
   \[
   \Delta = (3y + 3)^2 - 4(y - 1)(4y - 4) \geq 0.
   \]
   
   Simplify the discriminant:
   
   \[
   (3y + 3)^2 - 4(y - 1)(4y - 4) = 9y^2 + 18y + 9 - 16y^2 + 32y - 16.
   \]
   
   Combine like terms:
   
   \[
   -7y^2 + 50y - 7 \geq 0.
   \]
   
5. **Solve the Inequality for \(y\):**
   Solving the above inequality will give the range of \(y\).

The resulting values of \(y\) from solving the above inequality provide the range of the function.

Would you like further details on any of these steps, or do you have any questions?

### Related Questions:

1. How can you solve the inequality \(-7y^2 + 50y - 7 \geq 0\) to find the exact range of \(y\)?
2. What is the significance of the discriminant in determining the range of values?
3. How does completing the square help in finding the range of rational functions?
4. Can you find the critical points of the function and analyze their impact on the range?
5. What happens to the function as \(x\) approaches positive or negative infinity?

### Tip:
When finding the range of a rational function, always consider the critical points, asymptotes, and the behavior at infinity to get a complete understanding of its values.

Discover how to find the range of the rational function y = (x^2 - 3x + 4) / (x^2 + 3x + 4) with detailed steps and solutions. Explore the behavior of the function, including the use of the discriminant in determining the domain restrictions. Suitable for students in Grades 11-12.

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