To determine the range of the function \( f(x) = \frac{x}{{x^2 - x - 6}} \), we need to follow a few steps:

### 1. Factor the Denominator
The first thing to do is factor the denominator:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

So the function can be rewritten as:

\[
f(x) = \frac{x}{(x - 3)(x + 2)}
\]

### 2. Identify the Domain
The domain of the function excludes the points where the denominator is zero. Therefore, we solve:

\[
(x - 3)(x + 2) = 0
\]

This gives \( x = 3 \) and \( x = -2 \), which are vertical asymptotes. The domain is \( x \in (-\infty, -2) \cup (-2, 3) \cup (3, \infty) \).

### 3. Analyze Behavior at Asymptotes and Limits
To understand the range, we also consider the behavior near the vertical asymptotes \( x = 3 \) and \( x = -2 \), and as \( x \to \infty \) and \( x \to -\infty \).

- As \( x \to 3^+ \) or \( x \to 3^- \), \( f(x) \to \infty \) or \( f(x) \to -\infty \).
- As \( x \to -2^+ \) or \( x \to -2^- \), \( f(x) \to \infty \) or \( f(x) \to -\infty \).
- As \( x \to \infty \), the function tends to 0, i.e., \( \lim_{x \to \infty} f(x) = 0 \).
- As \( x \to -\infty \), the function also tends to 0, i.e., \( \lim_{x \to -\infty} f(x) = 0 \).

However, the function never actually reaches \( f(x) = 0 \).

### 4. Conclusion About the Range
The function can take all real values except 0. Therefore, the range of the function is:

\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\]

Would you like further details on any specific part of this process?

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Here are 5 related questions for further exploration:
1. What is the domain of other rational functions?
2. How do you find vertical asymptotes for different functions?
3. How does the behavior of \( f(x) \) change near its horizontal asymptotes?
4. What is the relationship between the degree of the numerator and the denominator for the range of rational functions?
5. How can we determine horizontal asymptotes for more complex functions?

**Tip:** When analyzing rational functions, factoring the denominator and numerator can simplify identifying asymptotes and domain restrictions.

Explore the steps to find the range of the rational function f(x) = x / (x^2 - x - 6), including factorization, identifying vertical asymptotes, and analyzing behavior at limits. Suitable for pre-calculus and calculus students in Grades 11-12.

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