Ohio Assessments for Educators (OAE) Mathematics Practice Exam

In the context of rational functions, what happens if the degree of the numerator is greater than the degree of the denominator by exactly one?

1. There will be a horizontal asymptote

2. There will be an oblique or diagonal asymptote

3. The function will have no asymptotes

4. The function will approach a vertical line

The correct answer is: There will be an oblique or diagonal asymptote

When analyzing rational functions, the relationship between the degrees of the numerator and the denominator plays a crucial role in determining the behavior of the function, particularly regarding asymptotes. If the degree of the numerator is exactly one greater than the degree of the denominator, the rational function will indeed have an oblique (or diagonal) asymptote. This occurs because, when performing polynomial long division, the result will yield a linear expression plus a remainder term. The linear expression obtained from the long division represents the oblique asymptote. As the value of \( x \) approaches infinity or negative infinity, the remainder becomes negligible, and the function behaves similarly to the linear expression, thus forming a diagonal asymptote. The presence of an oblique asymptote suggests that the function does not settle into a horizontal approach, which distinguishes it from situations where the degrees are equal (resulting in a horizontal asymptote) or when the degree of the numerator is less than that of the denominator (which might lead to a horizontal asymptote at zero). Hence, understanding the relationship between polynomial degrees in rational functions is essential for identifying the type of asymptotes present.