Question

Which of the following functions has a horizontal asymptote at y = 3?

1) A rational function increasing from the left to right never touching x equals -3 and y equals 2.
2) A rational function increasing from left never touching x equals -3 and going to a maximum of the coordinate (0,0) and then decreasing to the right and never touching x equals 3.
3) A rational function decreasing from the left to right never touching x equals 3 and y equals 2.
4) A rational function increasing from the left to right never touching x equals -2 and y equals 3.
5) A rational function increasing from left never touching x equals -2 and going to a maximum of the coordinate (0,0) and then decreasing to the right and never touching x equals 2.
6) A rational function decreasing from the left to right never touching x equals 2 and y equals 3.
7) A rational function increasing from the left to right never touching x equals -3 and y equals 4.
8) A rational function increasing from left never touching x equals -3 and going to a maximum of the coordinate (0,0) and then decreasing to the right and never touching x equals 3.
9) A rational function decreasing from the left to right never touching x equals 3 and y equals 4.
10) A rational function increasing from the left to right never touching x equals -2 and y equals 1.
11) A rational function increasing from left never touching x equals -2 and going to a maximum of the coordinate (0,0) and then decreasing to the right and never touching x equals 2.
12) A rational function decreasing from the left to right never touching x equals 2 and y equals 1.

Answer 1

Option 4, 'A rational function increasing from the left to right never touching x equals -2 and y equals 3,' is the function that has a horizontal asymptote at y = 3, as the function approaches this value but never reaches it.

Step-by-step explanation:

To find which of the given rational functions has a horizontal asymptote at y = 3, we should look for a function where the value of y approaches 3 as x approaches infinity (increases without bound) or negative infinity (decreases without bound). The asymptotes are lines that the graph of a function approaches but never actually touches, which is a characteristic behavior of rational functions.Among the options given, option 4 states 'A rational function increasing from the left to right never touching x equals -2 and y equals 3'. This indicates that as x increases or decreases, the function approaches but does not touch the line y=3. Therefore, it suggests the presence of a horizontal asymptote at y = 3, which is what we are looking for.