Question

use long division to rewrite the rational function. what are the asymptotes of f? sketch the graph.
$f(x) = \frac{4x}{x + 6}$

$f(x) = 4 - \frac{24}{x + 6}$

what are the horizontal asymptotes of f? select the correct choice below and, if necessary, fill in the answer box within your choice.
a. there is a horizontal asymptote defined by the line $y = 4$.
( type an equation. type an exact answer, using radicals as needed. )
b. there is no horizontal asymptote.

what are the vertical asymptotes of f? select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
a. there are two vertical asymptotes. the left - most asymptote is the line \\(\square\\) and the right - most asymptote is the line \\(\square\\).
( type equations. type exact answers, using radicals as needed. )
b. there is a single vertical asymptote. the vertical asymptote is the line \\(\square\\).
( type an equation. type an exact answer, using radicals as needed. )
c. there is no vertical asymptote.

Explanation:

For Horizontal Asymptote:

The function is rewritten as \( f(x) = 4 - \frac{24}{x + 6} \). For a rational function, when the degrees of the numerator and denominator are equal (or in this case, as \( x \to \pm\infty \), the term \( \frac{24}{x + 6} \to 0 \)), we look at the ratio of the leading coefficients. Here, the horizontal asymptote is determined by the constant term when the fractional part approaches zero. As \( x \) becomes very large (positive or negative), \( \frac{24}{x + 6} \) approaches 0, so \( f(x) \) approaches 4. Thus, the horizontal asymptote is \( y = 4 \), so option A is correct.

For Vertical Asymptote:

Vertical asymptotes occur where the denominator of the rational function is zero (and the numerator is not zero at that point). The original function is \( f(x)=\frac{4x}{x + 6} \), and the denominator is \( x + 6 \). Setting the denominator equal to zero: \( x + 6 = 0 \), which gives \( x=-6 \). We check the numerator at \( x = -6 \): the numerator is \( 4(-6)=-24
eq0 \), so there is a single vertical asymptote at \( x=-6 \). So we choose option B, and the vertical asymptote is the line \( x = -6 \).

Horizontal Asymptote Answer:

A. There is a horizontal asymptote defined by the line \( y = 4 \).

Vertical Asymptote Answer:

B. There is a single vertical asymptote. The vertical asymptote is the line \( x = -6 \).

Answer:

For Horizontal Asymptote:

The function is rewritten as \( f(x) = 4 - \frac{24}{x + 6} \). For a rational function, when the degrees of the numerator and denominator are equal (or in this case, as \( x \to \pm\infty \), the term \( \frac{24}{x + 6} \to 0 \)), we look at the ratio of the leading coefficients. Here, the horizontal asymptote is determined by the constant term when the fractional part approaches zero. As \( x \) becomes very large (positive or negative), \( \frac{24}{x + 6} \) approaches 0, so \( f(x) \) approaches 4. Thus, the horizontal asymptote is \( y = 4 \), so option A is correct.

For Vertical Asymptote:

Vertical asymptotes occur where the denominator of the rational function is zero (and the numerator is not zero at that point). The original function is \( f(x)=\frac{4x}{x + 6} \), and the denominator is \( x + 6 \). Setting the denominator equal to zero: \( x + 6 = 0 \), which gives \( x=-6 \). We check the numerator at \( x = -6 \): the numerator is \( 4(-6)=-24
eq0 \), so there is a single vertical asymptote at \( x=-6 \). So we choose option B, and the vertical asymptote is the line \( x = -6 \).

Horizontal Asymptote Answer:

A. There is a horizontal asymptote defined by the line \( y = 4 \).

Vertical Asymptote Answer:

B. There is a single vertical asymptote. The vertical asymptote is the line \( x = -6 \).