Understanding Domains and Asymptotes in Functions

The video tutorial explains how to determine the domain of rational functions, focusing on functions like f(x) = 1/(x^2 - 4), f(x) = 1/(x^2 - 9), and f(x) = 1/x^2. It highlights the importance of ensuring the denominator is not zero and uses graphical representations to illustrate vertical asymptotes. The tutorial also covers how to express domains using interval notation and emphasizes understanding the exclusion of specific x-values that make the denominator zero.

What values of x make the denominator zero in the function f(x) = 1/(x^2 - 4)?

• x = 4

• x = 0

• x = -4

• x = -2 and x = 2

Why does the graph of f(x) = 1/(x^2 - 4) not touch the vertical lines at x = -2 and x = 2?

• Because they are vertical asymptotes

• Because they are horizontal asymptotes

• Because the graph is a circle

• Because the graph is a straight line

What is the significance of vertical asymptotes in a graph?

• They indicate where the graph approaches but never touches

• They indicate where the graph touches the x-axis

• They indicate where the graph touches the y-axis

• They indicate where the graph is a straight line

Which of the following is true about the function f(x) = 1/(x^2 - 4)?

• It is a quadratic function

• It is a linear function

• It has breaks in the graph

• It is a polynomial

For the function f(x) = 1/(x^2 - 9), which values of x are not allowed?

• x = 0

• x = 9

• x = -9

• x = -3 and x = 3

What happens to the graph of f(x) = 1/(x^2 - 9) at x = 3?

• It touches the line

• It crosses the line

• It approaches but never touches the line

• It becomes a straight line

What is the domain of the function y = 1/x^2?

• All positive numbers

• All negative numbers

• All real numbers

• All real numbers except 0

How can the domain of y = 1/x^2 be expressed using interval notation?

• (-∞, ∞)

• [0, ∞)

• (-∞, 0) ∪ (0, ∞)

• (-∞, 0] ∪ [0, ∞)

In the function f(x) = 1/(x^2 - 9), how can the domain be expressed using interval notation?

• (-∞, 3) ∪ (3, ∞)

• (-∞, ∞)

• (-3, 3)

• (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)

How can the domain of a function be described?

• All possible y-values

• All possible x-values

• All possible w-values

• All possible z-values