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The given function is:

\[
y = \frac{1}{x^2 + 4x + 7} - 3
\]

### Explanation:

1. **Function Analysis**:
   - The function consists of a rational term \(\frac{1}{x^2 + 4x + 7}\) and a constant term \(-3\).
   - The denominator of the rational term \(x^2 + 4x + 7\) will determine where the function is undefined.

2. **Domain**:
   The denominator \(x^2 + 4x + 7\) must not equal zero. Solve:
   \[
   x^2 + 4x + 7 = 0
   \]
   The discriminant (\(D\)) is:
   \[
   D = b^2 - 4ac = 4^2 - 4(1)(7) = 16 - 28 = -12
   \]
   Since \(D < 0\), the quadratic has no real roots. Therefore, \(x^2 + 4x + 7 \neq 0\) for all real \(x\), and the function is defined for all real numbers.

3. **Range**:
   - The term \(\frac{1}{x^2 + 4x + 7}\) is always positive because \(x^2 + 4x + 7 > 0\) for all real \(x\).
   - The range of \(\frac{1}{x^2 + 4x + 7}\) lies in \((0, 1/7]\), since the minimum value of \(x^2 + 4x + 7\) is 7 (achieved at the vertex \(x = -2\)).
   - Subtracting 3 shifts the range down to \((-3, -3 + \frac{1}{7}]\).

4. **Key Points**:
   - Vertex of \(x^2 + 4x + 7\) occurs at \(x = -2\).
   - At \(x = -2\), the maximum value of \(y\) is:
     \[
     y = \frac{1}{7} - 3 = -\frac{20}{7}.
     \]
   - As \(x \to \pm \infty\), \(y \to -3\).

### Summary:
- **Domain**: \(x \in \mathbb{R}\)
- **Range**: \((-3, -\frac{20}{7}]\)

Would you like further details or a graph of this function? 

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Here are some related questions to deepen your understanding:
1. How do you determine the vertex of a quadratic function?
2. What does the discriminant tell us about a quadratic equation?
3. How do horizontal asymptotes influence the behavior of rational functions?
4. Can you explore how shifts in the numerator or denominator affect rational functions?
5. What happens to the range if we multiply the rational term by a constant?

**Tip**: Always examine the discriminant of a quadratic expression to understand if the function has real roots.

Graph the function y = 1 / (x^2 + 4x + 7) - 3.

Explore the graph of y = 1 / (x^2 + 4x + 7) - 3, including its domain, range, and asymptotic behavior. Understand the role of quadratic expressions in shaping rational functions. Suitable for Grades 9-11 students studying algebra and graphing.

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