To solve the given problem, we need to find the horizontal asymptotes of the function:

$f(x) = \frac{(3x - 5)(x + 7)}{3(3x + 4)(x + 7)}.$

Step 1: Simplify the function if possible

First, observe that the factor $(x + 7)$ appears in both the numerator and denominator. As long as $x \neq -7$, these terms cancel out:

$f(x) = \frac{3x - 5}{3(3x + 4)} \quad \text{for } x \neq -7.$

So, the simplified function becomes:

$f(x) = \frac{3x - 5}{9x + 12}.$

Step 2: Determine horizontal asymptotes

Horizontal asymptotes depend on the degrees of the numerator and denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In this case:

• The numerator $3x - 5$ has degree 1.
• The denominator $9x + 12$ also has degree 1.

Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients:

$y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{3}{9} = \frac{1}{3}.$

Final Answer:

The horizontal asymptote of the function is:

$y = \frac{1}{3}.$

Let me know if you'd like further details or have additional questions!

Here are 5 related questions to consider:

1. How do vertical asymptotes differ from horizontal asymptotes?
2. What happens to the graph of $f(x)$ near $x = -7$?
3. Can a function have more than one horizontal asymptote?
4. How would a slant asymptote be identified in a similar problem?
5. What are the restrictions on the domain of $f(x)$ in this problem?

Tip: Always simplify a rational function fully before analyzing asymptotes to ensure accuracy.