Discontinuities and Domains in Functions

The video tutorial explains rational functions, focusing on their definition, domain, and graph characteristics. It distinguishes between continuous and discontinuous graphs, highlighting points of discontinuity. The tutorial further explores removable and non-removable discontinuities, providing examples and methods to identify and classify them. It concludes with practical exercises on finding points of discontinuity and determining x and y intercepts.

What is a rational function?

• A function that is always continuous.

• A function that is always discontinuous.

• A function that can be expressed as a ratio of two polynomials.

• A function that has no intercepts.

Which of the following is true about the domain of a rational function?

• It includes all real numbers.

• It excludes values that make the denominator zero.

• It excludes values that make the numerator zero.

• It includes only positive numbers.

What characterizes a continuous graph?

• It has jumps and breaks.

• It is always a straight line.

• It has holes.

• It can be drawn without lifting the pencil.

In a discontinuous graph, what does a hole represent?

• A point where the function is negative.

• A point where the function is infinite.

• A point where the function is zero.

• A point where the function is undefined.

What is a removable discontinuity?

• A discontinuity that is always at x = 0.

• A discontinuity that cannot be removed.

• A discontinuity that can be removed by redefining the function.

• A discontinuity that occurs at infinity.

How can a removable discontinuity be addressed?

• By changing the variable.

• By adding more terms to the function.

• By redefining the function at that point.

• By ignoring it.

What is a non-removable discontinuity?

• A discontinuity that occurs only at x = 0.

• A discontinuity that represents a break in the graph.

• A discontinuity that is always removable.

• A discontinuity that can be removed by redefining the function.

In the function y = (x + 3)(x + 2)/(x + 2), what type of discontinuity is at x = -2?

• No discontinuity

• Infinite discontinuity

• Removable discontinuity

• Non-removable discontinuity

What is the domain of the function y = x + 3 / (x^2 - 4x + 3)?

• All real numbers

• All real numbers except x = 1 and x = 3

• All real numbers except x = 0

• All real numbers except x = -3

What are the x-intercepts of the function y = x + 3 / (x^2 - 4x + 3)?

• x = 3

• x = -3

• x = 1

• x = 0