Question

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

rewrite the function using long division.
$f(x)=\square$

Explanation:

Step1: Divide 4x by x

We divide the numerator \(4x\) by the first term of the denominator \(x\). So, \(\frac{4x}{x} = 4\).

Step2: Multiply (x + 6) by 4

Multiply the entire denominator \(x + 6\) by 4, we get \(4(x + 6)=4x + 24\).

Step3: Subtract and find the remainder

Subtract \(4x + 24\) from \(4x\) (the original numerator). So, \(4x-(4x + 24)=4x - 4x - 24=-24\).

Step4: Rewrite the function

A rational function \(\frac{a(x)}{b(x)}\) can be written as quotient \(+\frac{\text{remainder}}{\text{divisor}}\) when using long division. Here, the quotient is 4, the remainder is -24, and the divisor is \(x + 6\). So, \(\frac{4x}{x + 6}=4+\frac{-24}{x + 6}=4-\frac{24}{x + 6}\).

Answer:

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