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)=\square

Explanation:

Step1: Perform Long Division

We divide \(3x\) by \(x + 4\). We know that \(3(x + 4)=3x+12\), so we can write \(3x = 3(x + 4)-12\). Then, \(\frac{3x}{x + 4}=\frac{3(x + 4)-12}{x + 4}\).

Step2: Simplify the Expression

We split the fraction: \(\frac{3(x + 4)-12}{x + 4}=\frac{3(x + 4)}{x + 4}-\frac{12}{x + 4}\). Since \(x
eq - 4\) (to avoid division by zero), \(\frac{3(x + 4)}{x + 4}=3\). So the function becomes \(f(x)=3-\frac{12}{x + 4}\).

Answer:

\(3-\dfrac{12}{x + 4}\)