To determine the vertical, horizontal, and oblique asymptotes of the rational function $f(x) = \frac{6 - 4x}{6x + 5}$, we will analyze the function in the following steps:

1. Vertical Asymptotes:

Vertical asymptotes occur where the denominator of the rational function is equal to zero (assuming the numerator is not also zero at that point, which would indicate a hole rather than an asymptote). So, let's find where the denominator is zero:

$6x + 5 = 0$ Solving for $x$:

$6x = -5 \quad \Rightarrow \quad x = -\frac{5}{6}$

Thus, there is a vertical asymptote at $x = -\frac{5}{6}$.

2. Horizontal Asymptotes:

Horizontal asymptotes depend on the degree of the polynomial in the numerator and the denominator.

• The degree of the numerator $6 - 4x$ is 1 (since the highest power of $x$ is 1).
• The degree of the denominator $6x + 5$ is also 1 (the highest power of $x$ is 1).

When the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients. Here, the leading coefficient of the numerator is $-4$ (from $-4x$), and the leading coefficient of the denominator is $6$ (from $6x$).

Thus, the horizontal asymptote is:

$y = \frac{-4}{6} = -\frac{2}{3}$

So, there is a horizontal asymptote at $y = -\frac{2}{3}$.

3. Oblique (Slant) Asymptotes:

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, both the numerator and the denominator have the same degree (both are degree 1), so there is no oblique asymptote for this function.

Summary of Asymptotes:

• Vertical Asymptote: $x = -\frac{5}{6}$
• Horizontal Asymptote: $y = -\frac{2}{3}$
• Oblique Asymptote: None

Would you like further clarification on any of these concepts? Or do you have more questions?

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Additional questions:

1. How do you find the domain of a rational function?
2. Can a rational function have both a vertical and horizontal asymptote?
3. What happens when the numerator and denominator have different degrees?
4. How do you graph rational functions with asymptotes?
5. What is the relationship between holes and vertical asymptotes in rational functions?

Tip: Always check the degrees of the numerator and denominator to determine the type of asymptote (vertical, horizontal, or oblique).