To find the range of the function \( y = \frac{x^2 - 2x - 3}{x + 4} \), let's analyze it step by step:

### Step 1: Simplify the expression
Factor the quadratic expression in the numerator:
\[
x^2 - 2x - 3 = (x - 3)(x + 1)
\]
So the function becomes:
\[
y = \frac{(x - 3)(x + 1)}{x + 4}
\]

### Step 2: Identify discontinuities
The function will be undefined where the denominator equals zero. Setting \( x + 4 = 0 \) gives:
\[
x = -4
\]
Thus, \( x = -4 \) is a vertical asymptote.

### Step 3: Analyze end behavior (asymptotes)
As \( x \to \infty \) or \( x \to -\infty \), the degree of the numerator is larger than the degree of the denominator. Hence, the function will tend to infinity as \( x \) grows larger or smaller.

### Step 4: Check for horizontal asymptotes or limits at infinity
There is no horizontal asymptote because the degree of the numerator (2) is greater than the degree of the denominator (1). Instead, there will be a slant (oblique) asymptote. To find it, we perform polynomial long division:

Performing the division:
\[
\frac{x^2 - 2x - 3}{x + 4} = x - 6 + \frac{21}{x+4}
\]
So, as \( x \to \infty \), the function behaves like \( y = x - 6 \). This is the slant asymptote.

### Step 5: Consider the behavior near the vertical asymptote
As \( x \to -4^+ \) (from the right), \( y \to +\infty \).
As \( x \to -4^- \) (from the left), \( y \to -\infty \).

### Step 6: Find critical points
Now, let's check the values of \( y \) at other critical points. Set \( x = 0 \) to find:
\[
y = \frac{0^2 - 2(0) - 3}{0 + 4} = \frac{-3}{4} = -\frac{3}{4}
\]

Similarly, you can check other specific points to understand how the function behaves around the vertical asymptote.

### Conclusion: Range
Based on the analysis:
- The function can take all real values because the slant asymptote allows the function to approach any real number as \( x \to \infty \) or \( x \to -\infty \), and there's no horizontal asymptote restricting the range.
- Therefore, the **range of the function is all real numbers** \( y \in (-\infty, \infty) \), except for possible exclusions due to local behavior.

Would you like me to go into more detail on any part of this?

Here are 5 related questions:
1. How do you find the vertical asymptotes of a rational function?
2. What is the significance of a slant asymptote?
3. How does the degree of the numerator and denominator affect a rational function's behavior?
4. How can you perform polynomial long division to find slant asymptotes?
5. How would you graph a function like this step by step?

**Tip:** When dealing with rational functions, always check for both vertical and horizontal asymptotes to better understand the function's behavior at infinity.

Explore how to find the range of the rational function y = (x^2 - 2x - 3)/(x + 4). Learn about vertical and slant asymptotes, polynomial long division, and quadratic factorization. This detailed guide is ideal for students in Grades 10-12 studying rational functions and asymptotes.

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