A rational function is a function that can be written as a fraction whose numerator and denominator are polynomials. However, when writing our function, we will focus only on the rational functions that can be written in the following form.
y= ax- b+ c
A function in the above form has a verticalasymptote at x= b and a horizontal asymptote at y= c. Therefore, to obtain a function with a vertical asymptote at x= - 2 and a horizontal asymptote at y= 4, we should let b be equal to - 2 and let c be 4.
y= ax-( - 2)+ 4
⇕
y=a/x+2+4
Finally, we have to arbitrarily choose the value of a. It can be any real number different from 0. Note that if a was equal to 0, the function would simplify to a linear function. We will let a be equal to 3.
y=3/x+2+4
Below you can see the graph of our function with its asymptotes marked.
Please note that our answer is one of infinitely many rational functions with a vertical asymptote at x=- 2 and a horizontal asymptote at y=4. Some of the ways that we can obtain other examples of such rational functions are by choosing a different constant for the numerator or by multiplying the denominator by a polynomial.
