Question

part 3 of 4
use long division to rewrite the rational function. what are the asymptotes of f? sketch the graph.
f(x) = \frac{3x}{x + 4}

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.
a. there is a horizontal asymptote defined by the line y = 3.
(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 is a single vertical asymptote. the vertical asymptote is the line \square
(type an equation. type an exact answer, using radicals as needed.)
b. 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.)
c. there is no vertical asymptote.

Explanation:

For Horizontal Asymptote:

To find the horizontal asymptote of a rational function \( f(x)=\frac{3x}{x + 4}\) (or its rewritten form \( f(x)=3-\frac{12}{x + 4}\)), we analyze the degrees of the numerator and the denominator.

• The degree of the numerator (for the original rational function form) is \(1\) (since the highest power of \(x\) in the numerator \(3x\) is \(1\)) and the degree of the denominator \(x + 4\) is also \(1\).
• When the degrees of the numerator and the denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is \(3\) and the leading coefficient of the denominator is \(1\). So, the horizontal asymptote is \(y=\frac{3}{1}=3\). So, option A is correct with \(y = 3\).

For Vertical Asymptote:

To find the vertical asymptote of a rational function \( f(x)=\frac{3x}{x + 4}\), we set the denominator equal to zero and solve for \(x\) (since the function is undefined when the denominator is zero, and as \(x\) approaches the value that makes the denominator zero, the function tends to \(\pm\infty\)).

• Set the denominator \(x + 4=0\).
• Solving for \(x\), we get \(x=- 4\).
• So, there is a single vertical asymptote, and the equation of the vertical asymptote is \(x=-4\). So, for the vertical asymptote question, we choose option A and fill in \(x=-4\).

Final Answers:

• Horizontal Asymptote: A. There is a horizontal asymptote defined by the line \( y = 3\).
• Vertical Asymptote: A. There is a single vertical asymptote. The vertical asymptote is the line \( x=-4\).

Answer:

For Horizontal Asymptote:

To find the horizontal asymptote of a rational function \( f(x)=\frac{3x}{x + 4}\) (or its rewritten form \( f(x)=3-\frac{12}{x + 4}\)), we analyze the degrees of the numerator and the denominator.

• The degree of the numerator (for the original rational function form) is \(1\) (since the highest power of \(x\) in the numerator \(3x\) is \(1\)) and the degree of the denominator \(x + 4\) is also \(1\).
• When the degrees of the numerator and the denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is \(3\) and the leading coefficient of the denominator is \(1\). So, the horizontal asymptote is \(y=\frac{3}{1}=3\). So, option A is correct with \(y = 3\).

For Vertical Asymptote:

To find the vertical asymptote of a rational function \( f(x)=\frac{3x}{x + 4}\), we set the denominator equal to zero and solve for \(x\) (since the function is undefined when the denominator is zero, and as \(x\) approaches the value that makes the denominator zero, the function tends to \(\pm\infty\)).

• Set the denominator \(x + 4=0\).
• Solving for \(x\), we get \(x=- 4\).
• So, there is a single vertical asymptote, and the equation of the vertical asymptote is \(x=-4\). So, for the vertical asymptote question, we choose option A and fill in \(x=-4\).

Final Answers:

• Horizontal Asymptote: A. There is a horizontal asymptote defined by the line \( y = 3\).
• Vertical Asymptote: A. There is a single vertical asymptote. The vertical asymptote is the line \( x=-4\).