Question

Give an example of a rational function that has a horizontal asymptote of y = 2/9.

Answer 1

Note that the graph of the function $\displaystyle{ f(x)= \frac{1}{x}$ has 2 asymptotes:

The horizontal asymptote x=0, 

and the vertical asymptote y=0, as shown in the first picture.


Adding 2/9 to this function, creating $\displaystyle{ g(x)= \frac{1}{x}+\frac{2}{9}$, shifts the first graph 2/9 units up. It also shifts the horizontal asymptote y=0 to t=2/9.

We can express the function as $\displaystyle{ \frac{1}{x}+ \frac{2}{9}= \frac{9}{9x}+ \frac{2x}{9x}= \frac{2x+9}{9x}$.


Answer: $\displaystyle{ \frac{2x+9}{9x}$