Understanding Domain and Range of Rational Functions

The video tutorial explains how to determine the domain and range of the function f(x) = 1/(x + 2). It covers two main methods: graphical analysis and reasoning based on the properties of rational functions. The tutorial highlights the importance of identifying breaks in the graph, such as at x = -2, which affects the domain. It also discusses how to express domain and range using interval notation. The video concludes by emphasizing that the range excludes y = 0, as the function cannot equal zero when the numerator is a constant.

What is the domain of the function f(x) = 1/(x + 2)?

• All real numbers except x = 0

• All real numbers except x = 2

• All real numbers

• All real numbers except x = -2

Which of the following methods can be used to determine the domain and range of a function?

• Graphical analysis

• Algebraic manipulation

• Numerical approximation

• Statistical analysis

Why is there a break in the graph of f(x) = 1/(x + 2) at x = -2?

• Because x = -2 is a root of the function

• Because x = -2 is a maximum point

• Because x = -2 makes the denominator zero

• Because x = -2 is an asymptote

How is the domain of f(x) = 1/(x + 2) expressed in interval notation?

• (-∞, 2) ∪ (2, ∞)

• (-∞, -2) ∪ (-2, ∞)

• (-∞, ∞)

• (-∞, 0) ∪ (0, ∞)

What is the range of the function f(x) = 1/(x + 2)?

• All real numbers except y = -1

• All real numbers

• All real numbers except y = 1

• All real numbers except y = 0

In interval notation, how is the range of f(x) = 1/(x + 2) expressed?

• (-∞, ∞)

• (-∞, -1) ∪ (-1, ∞)

• (-∞, 1) ∪ (1, ∞)

• (-∞, 0) ∪ (0, ∞)

What property of rational functions helps determine the domain?

• The function must be continuous

• The denominator must not be zero

• The numerator must be zero

• The function must have a maximum

Why can't the fraction 1/(x + 2) be equal to zero?

• Because the numerator is a constant

• Because the function is continuous

• Because the denominator is a variable

• Because the function is undefined

What happens to the y-values as x approaches -2 from the right?

• They approach positive infinity

• They remain constant

• They approach zero

• They approach negative infinity

What is the significance of the constant numerator in the function f(x) = 1/(x + 2)?

• It causes the function to have a maximum

• It ensures the function is always positive

• It prevents the function from being zero

• It makes the function linear