To find the range of the function \( f(x) = \frac{x^2 + 2x + 1}{x^2 - x - 6} \), we'll analyze the behavior of the function by looking at its critical points, asymptotes, and the overall structure of the function.

### Step 1: Factor the numerator and denominator
First, let's factor both the numerator and the denominator where possible.

- **Numerator**: \( x^2 + 2x + 1 = (x + 1)^2 \)
- **Denominator**: \( x^2 - x - 6 \)

The denominator can be factored as:
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

So the function can be rewritten as:
\[
f(x) = \frac{(x + 1)^2}{(x - 3)(x + 2)}
\]

### Step 2: Identify Vertical Asymptotes and Discontinuities
The function has vertical asymptotes or points of discontinuity where the denominator is zero. This occurs at \( x = 3 \) and \( x = -2 \).

### Step 3: Analyze Behavior at Infinity
To determine the behavior of \( f(x) \) as \( x \) approaches infinity or negative infinity, we consider the degrees of the polynomial in the numerator and denominator. Both are degree 2 polynomials.

At large values of \( x \), the function behaves like:
\[
f(x) \approx \frac{x^2}{x^2} = 1
\]

### Step 4: Find Critical Points
To find the critical points, take the derivative of \( f(x) \) and set it equal to zero.

However, analyzing the simplified form:
\[
f(x) = 1 + \frac{2x + 7}{(x - 3)(x + 2)}
\]
we can see that the function equals 1 when \( \frac{2x + 7}{(x - 3)(x + 2)} = 0 \), which never happens because the numerator 2x + 7 is non-zero for all x.

### Step 5: Determine Limits at the Asymptotes
Let's check the limits as \( x \) approaches the asymptotes:
\[
\lim_{x \to 3^{+}} f(x) = +\infty, \quad \lim_{x \to 3^{-}} f(x) = -\infty
\]
\[
\lim_{x \to -2^{+}} f(x) = -\infty, \quad \lim_{x \to -2^{-}} f(x) = +\infty
\]

### Step 6: Conclusion
Given that the function approaches infinity or negative infinity near the vertical asymptotes and approaches 1 as \( x \) approaches infinity, we conclude that the function's range is:

\[
\boxed{(-\infty, 1) \cup (1, +\infty)}
\]

This range excludes 1, as the function never actually takes the value 1.

Would you like more details or have any questions? 

Here are five related questions:
1. How do you find the domain of a rational function?
2. What are horizontal and vertical asymptotes, and how are they determined?
3. How can you use derivatives to find the critical points of a function?
4. What is the importance of factoring polynomials in rational functions?
5. How do you analyze the end behavior of a rational function?

**Tip:** Always check for discontinuities (like vertical asymptotes) when analyzing the range of a rational function.

Learn how to find the range of the rational function f(x)=x^2+2x+1/x^2-x-6 with a detailed step-by-step solution. This problem covers key concepts in rational functions, vertical asymptotes, and limit analysis. Suitable for high school students.

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