Question

use long division to rewrite the rational function. what are the asymptotes of f? sketch the graph. f(x)=\frac{3x}{x + 4} rewrite the function using long division. f(x)=3 - \frac{12}{x + 4} what are the horizontal asymptotes of f? select the correct choice below and, if necessary, fill in the answer box within your choice. \bigcirc a. there is a horizontal asymptote defined by the line \square. (type an equation. type an exact answer, using radicals as needed.) \bigcirc b. there is no horizontal asymptote.

Explanation:

Step1: Recall horizontal asymptote rules

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator with degrees \( n \) and \( m \) respectively:

• If \( n < m \), horizontal asymptote is \( y = 0 \).
• If \( n = m \), horizontal asymptote is \( y=\frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).
• If \( n > m \), no horizontal asymptote (but may have oblique/slant asymptote).

For \( f(x)=\frac{3x}{x + 4} \) (or rewritten as \( 3-\frac{12}{x + 4} \)), the degree of numerator \( N(x)=3x \) is \( n = 1 \), degree of denominator \( D(x)=x + 4 \) is \( m = 1 \). So \( n=m \).

Step2: Calculate horizontal asymptote

Leading coefficient of \( N(x) \) is \( 3 \), leading coefficient of \( D(x) \) is \( 1 \). So horizontal asymptote is \( y=\frac{3}{1}=3 \).

Answer:

A. There is a horizontal asymptote defined by the line \( y = 3 \)