Question

1. write a rational function that has the end - behavior like y = 3x².

Explanation:

Step1: Recall the form of a rational function

A rational function is of the form $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. For the end - behavior of a rational function $\frac{P(x)}{Q(x)}$ to be like a polynomial $y = ax^n$, the degree of $P(x)$ must be $n$ more than the degree of $Q(x)$.

Step2: Determine the degrees of polynomials in the rational function

We want the end - behavior to be like $y = 3x^{2}$. So, we can choose $P(x)=3x^{3}$ and $Q(x)=x$.

Step3: Write the rational function

The rational function $f(x)=\frac{3x^{3}}{x}$ (simplifies to $3x^{2}$ for $x
eq0$). In general, we can write $f(x)=\frac{3x^{3}+bx^{2}+cx + d}{x + k}$ where $b,c,d,k$ are constants (and $x
eq - k$). A simple example is $f(x)=\frac{3x^{3}}{x}=3x^{2},x
eq0$. Another example could be $f(x)=\frac{3x^{3}+2x^{2}+x + 1}{x + 1}$.

Answer:

$f(x)=\frac{3x^{3}}{x}$ (or any rational function of the form $\frac{3x^{3}+bx^{2}+cx + d}{x + k}$ where $b,c,d,k$ are constants and $x
eq - k$)