Question

How do you find the vertical, horizontal or slant asymptotes for `g(x)=( -2x+3)/( 3x+1)`?

Answer 1

The VA is the value of x that makes the denominator =0 while the HA is coefficient of the x in the numerator divided by the coefficient of the x in the denominator.

Explanation:

The VA of a rational function is found by setting the denominator equal to zero and solving for x.
`3x+1=0`
`x= -1/3`

There are three rules for finding the HA. If the degree of the denominator is greater than the degree of the numerator, then the HA is x=0. If the degree of the numerator and denominator is the same, then the HA is x = coefficient of numerator divided by coefficient of denominator. If the degree of the numerator is greater, there is no HA. There would be a slant or oblique asymptote instead.

In this case, the degree of the numerator = the degree of the denominator, so the HA is `y=-2/3`.

This function does not have a slant asymptote. A slant asymptote occurs only when the degree of the numerator is greater than the degree of the denominator.

Note: "degree" means the highest exponent.