Unit 4C - Domain Restrictions & Discontinuities

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a from the left or right.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes are found by setting the denominator of the rational function equal to zero and solving for x.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b that a function approaches as x approaches infinity or negative infinity.

Front

How do you determine horizontal asymptotes for rational functions?

Back

For rational functions, compare the degrees of the numerator and denominator: If the degree of the numerator is less, y = 0; if equal, y = leading coefficient of numerator/leading coefficient of denominator; if greater, no horizontal asymptote.

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Front

How do you find slant asymptotes?

Back

To find a slant asymptote, perform polynomial long division on the rational function.

Front

What is a hole in a rational function?

Back

A hole occurs in a rational function at a point where both the numerator and denominator are zero, indicating a removable discontinuity.

Front

How do you identify holes in a rational function?

Back

To identify holes, factor the numerator and denominator, and find common factors that can be canceled.

Front

What does it mean for a function to be discontinuous?

Back

A function is discontinuous if there are points at which it is not defined or does not have a limit.

Front

What are the types of discontinuities?

Back

The main types of discontinuities are removable (holes), jump, and infinite (vertical asymptotes).