Question

Which key features are present in these three functions?

a) Horizontal asymptote: y = -2, Domain: (-[infinity], -3) ∪ (2, 6) ∪ (6, [infinity])
b) Oblique asymptote, Horizontal asymptote: y = 1, Domain: (-[infinity], 2) ∪ (2, 6) ∪ (6, [infinity])
c) Horizontal asymptote: y = -2, Domain: (-[infinity], -3) ∪ (2, 6) ∪ (6, [infinity])
d) Oblique asymptote, Horizontal asymptote: y = -2, Domain: (-[infinity], -3) ∪ (2, 6) ∪ (6, [infinity])

Answer 1

Key features of the described functions include horizontal asymptotes at y = -2 or y = 1, the presence of an oblique asymptote in some cases, and specific domains that indicate where the functions are defined without discontinuities.

Step-by-step explanation:

The key features present in these three functions include the presence of horizontal asymptotes, the domain in which the functions are defined, and for some, the presence of an oblique asymptote. A horizontal asymptote is a line that a function approaches as the independent variable goes to infinity. An oblique asymptote is a slanted line that a function approaches as the independent variable goes to infinity. Therefore:

• Function (a) and (c) both have a horizontal asymptote at y = -2 and identical domains.
• Function (b) is unique as it has both an oblique asymptote and a horizontal asymptote at y = 1.
• Function (d) has an oblique asymptote and a horizontal asymptote at y = -2.

When considering the domain for these functions, they avoid certain values where division by zero or other undefined behaviors occur, thus having intervals as specified in their domain descriptions. These intervals represent the x-values over which the function exists and avoids any discontinuities.