Question

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

Explanation:

Step1: Recall rational - function end - behavior rule

For a rational function $f(x)=\frac{a_nx^n + \cdots+a_0}{b_mx^m+\cdots + b_0}$, if $n = m$, the end - behavior is $y=\frac{a_n}{b_m}$; if $nm$, there is no horizontal asymptote. For the function $y=\frac{3}{x}$, the degree of the numerator is $0$ and the degree of the denominator is $1$.

Step2: Construct a rational function

A simple rational function with the same end - behavior as $y=\frac{3}{x}$ can be $f(x)=\frac{3}{x + 1}$ (the addition of a constant in the denominator does not change the end - behavior). In general, any function of the form $f(x)=\frac{3}{x + k}$ where $k$ is a real number will have the same end - behavior as $y=\frac{3}{x}$, because as $x\to\pm\infty$, the constant $k$ in the denominator becomes negligible.

Answer:

$f(x)=\frac{3}{x + 1}$ (or any function of the form $f(x)=\frac{3}{x + k},k\in\mathbb{R}$)