the horizontal asymptote for the graph of f(x)$$ \frac{9x}{3x-3} $$is

To find the horizontal asymptote for the function 

\[
f(x) = \frac{9x}{3x - 3}
\]

we should analyze the behavior of the function as $ x \to \infty $ and $ x \to -\infty $.

### Step 1: Simplify the Function

First, we can simplify $ f(x) $ by dividing both the numerator and the denominator by the highest power of $ x $, which in this case is $ x $:

\[
f(x) = \frac{9x}{3x - 3} = \frac{9}{3 - \frac{3}{x}}.
\]

As $ x \to \infty $ (and $ x \to -\infty $), the term $ \frac{3}{x} $ approaches $ 0 $. So, we are left with:

\[
f(x) \approx \frac{9}{3} = 3.
\]

### Conclusion

Thus, the horizontal asymptote of the function $ f(x) = \frac{9x}{3x - 3} $ is:

\[
y = 3.
\]

Would you like more details or have any questions?

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Here are five questions to deepen understanding:

1. How do horizontal asymptotes differ from vertical asymptotes?
2. What are the rules for finding horizontal asymptotes of rational functions?
3. How would the horizontal asymptote change if the degrees of the numerator and denominator were different?
4. What is the significance of the horizontal asymptote in real-world applications?
5. Can horizontal asymptotes intersect the graph of a function?

**Tip:** Horizontal asymptotes indicate the value that a function approaches as $ x $ becomes very large (positive or negative).

The horizontal asymptote for the graph of f(x) $$ \frac{9x}{3x-3} $$ is

Discover how to find the horizontal asymptote of the function f(x) = 9x/(3x-3). This guide includes a step-by-step analysis of the function as x approaches infinity, demonstrating key concepts in rational functions. Suitable for high school students in Grades 10-12.

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