Rational Functions - Discontinuity (VA & HA)

Front

What are vertical asymptotes?

Back

Vertical asymptotes are lines x = a where a rational function approaches infinity or negative infinity as the input approaches a. They occur where the denominator of the function is zero and the numerator is not zero.

Front

What are horizontal asymptotes?

Back

Horizontal asymptotes are lines y = b that a rational function approaches as x approaches infinity or negative infinity. They indicate the behavior of the function at extreme values of x.

Front

Define removable discontinuity.

Back

Removable discontinuity occurs when a rational function has a common factor in both the numerator and denominator, allowing the function to be redefined at that point.

Front

What is the significance of a point of discontinuity?

Back

A point of discontinuity indicates where a function is not continuous, which can occur due to removable discontinuities or vertical asymptotes.

Front

How do you find vertical asymptotes?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

Front

How do you find 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 the formula for finding horizontal asymptotes when degrees are equal?

Back

If the degrees of the numerator and denominator are equal, the horizontal asymptote is given by y = (leading coefficient of numerator) / (leading coefficient of denominator).

Front

What does it mean if a function has no horizontal asymptote?

Back

If a function has no horizontal asymptote, it means that as x approaches infinity or negative infinity, the function's values continue to increase or decrease without approaching a finite value.

Front

What is the relationship between discontinuities and asymptotes?

Back

Discontinuities in a rational function often correspond to vertical asymptotes, while horizontal asymptotes describe the end behavior of the function.

Front

What is an example of a rational function with a removable discontinuity?

Back

An example is f(x) = (x^2 - 1)/(x - 1). The function has a removable discontinuity at x = 1, where it can be simplified to f(x) = x + 1.