To find the horizontal and vertical asymptotes of the function:

$f(x) = \frac{3 - 7x}{3 + 2x}$

1. Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (since the function is undefined at these points).

Set the denominator equal to zero:

$3 + 2x = 0$

Solve for $x$:

$2x = -3 \quad \Rightarrow \quad x = -\frac{3}{2}$

Thus, the vertical asymptote is at: $x = -\frac{3}{2}$

2. Horizontal Asymptote

The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator.

• The degree of the numerator $3 - 7x$ is 1 (because of the $x$ term).
• The degree of the denominator $3 + 2x$ is also 1.

When the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is $-7$ and the leading coefficient of the denominator is $2$.

Thus, the horizontal asymptote is:

$y = \frac{-7}{2}$

Final Answer:

• Vertical Asymptote: $x = -\frac{3}{2}$
• Horizontal Asymptote: $y = \frac{-7}{2}$

Would you like further details or have any questions?

Relative questions:

1. How do you determine horizontal asymptotes when the degree of the numerator is greater than the denominator?
2. What happens if the degree of the denominator is greater than the numerator?
3. Can a function have more than one vertical asymptote?
4. How do you check for oblique asymptotes in rational functions?
5. What happens to the graph of a function near a vertical asymptote?

Tip:

To graph rational functions accurately, always determine asymptotes, intercepts, and behavior at infinity.