Questions: Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational fun f(x) = (3-3x)/(5x+3) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The equation of the vertical asymptote is x = . (Type an integer or a fraction. Simplify your answer.) B. There is no vertical asymptote. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The equation of the horizontal asymptote is y = 7. (Type an integer or a fraction. Simplify your answer.) B. There is no horizontal asymptote. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The equation of the oblique asymptote is y = . (Type an integer or a fraction. Simplify your answer.) B. There is no oblique asymptote.

Solution Steps

To find the vertical asymptote of the rational function \( f(x) = \frac{3 - 3x}{5x + 3} \), we set the denominator equal to zero:

\[ 5x + 3 = 0 \]

Solving for \( x \):

\[ 5x = -3 \implies x = -\frac{3}{5} \]

Thus, the equation of the vertical asymptote is \( x = -\frac{3}{5} \).

Next, we determine the horizontal asymptote by comparing the degrees of the numerator and denominator. The degree of the numerator \( (3 - 3x) \) is 1, and the degree of the denominator \( (5x + 3) \) is also 1. Since the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients:

\[ \text{Leading coefficient of the numerator} = -3, \quad \text{Leading coefficient of the denominator} = 5 \]

Thus, the horizontal asymptote is:

\[ y = \frac{-3}{5} \]

An oblique asymptote exists when the degree of the numerator is exactly one greater than the degree of the denominator. In this case, since the degree of the numerator is equal to the degree of the denominator, there is no oblique asymptote.

Final Answer