Question

1. write a rational function that has the end behavior like y = 3x.

Explanation:

Step1: Recall rational - function end - behavior rule

For a rational function $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, if the degree of $P(x)$ is one more than the degree of $Q(x)$, the function has a slant asymptote.

Step2: Construct the rational function

Let $P(x)=3x^{2}+x$ and $Q(x)=x$. Then the rational function $f(x)=\frac{3x^{2}+x}{x}$. Simplifying for $x
eq0$, we have $f(x) = 3x + 1$. As $x\to\pm\infty$, the end - behavior of $y = 3x+1$ is dominated by the term $3x$. Another example could be $f(x)=\frac{3x^{2}}{x}$ (for $x
eq0$), which simplifies to $y = 3x$ for $x
eq0$.

Answer:

$f(x)=\frac{3x^{2}}{x}$ (or $f(x)=\frac{3x^{2}+x}{x}$)