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.