Discontinuities of Rational Functions

The video tutorial covers the concept of discontinuities in functions, focusing on removable and non-removable types. It explains how to identify and differentiate between holes and vertical asymptotes using algebraic simplification of rational expressions. The tutorial includes several examples to illustrate these concepts and concludes with a discussion on continuous functions.

What is a removable point discontinuity commonly referred to as?

• A hole

• A jump

• A peak

• An asymptote

Which type of discontinuity is characterized by a graph approaching a value but never reaching it?

• Infinite discontinuity

• Jump discontinuity

• Removable discontinuity

• Oscillating discontinuity

When simplifying a rational expression, what indicates a vertical asymptote?

• A factor in the numerator

• A zero in the denominator that cannot be canceled

• A constant term

• A zero in the numerator

How can you determine if a discontinuity is removable?

• If the numerator and denominator have a common factor that can be canceled

• If the denominator is a constant

• If the numerator is zero

• If the expression is a polynomial

In the example where the numerator is factored as (x-2)(x+2), what type of discontinuity occurs at x=2?

• Vertical asymptote

• Removable discontinuity

• Jump discontinuity

• No discontinuity

What happens to the discontinuity when a factor in the denominator is canceled by a factor in the numerator?

• It becomes a vertical asymptote

• It becomes a jump

• It disappears completely

• It becomes a hole

In the example with the expression 2(x-3)(x+2), what are the discontinuities?

• x=0 and x=2

• x=5 and x=-5

• x=3 and x=-2

• x=1 and x=-1

What is the result when the denominator of a rational expression cannot be factored?

• The graph is continuous

• The expression is undefined

• There is a vertical asymptote

• There is a hole

What should you do if the denominator of a rational expression equals zero but cannot be solved under the real number system?

• Consider it a hole

• Consider the graph continuous

• Consider it a vertical asymptote

• Consider it a jump discontinuity

How can you verify the discontinuities and domain of a function graphically?

• By using a calculator

• By using a graphing tool like Desmos

• By solving the numerator

• By ignoring the denominator