Question

6.7.3 quiz: writing rational functions
question 3 of 10
an asymptote is a line that the graph of a function
a. divides
b. approaches and crosses
c. defines
d. equals
e. approaches but does not cross

Explanation:

An asymptote is a line that the graph of a function approaches but does not cross. This is a fundamental concept in calculus when studying the behavior of functions, especially rational functions. For example, in the function \(y = \frac{1}{x}\), the \(x\)-axis (\(y = 0\)) and \(y\)-axis (\(x=0\)) are asymptotes. As \(x\) gets very large or very small, the graph of \(y=\frac{1}{x}\) gets closer and closer to the \(x\)-axis (\(y = 0\)) but never touches it. Similarly, as \(x\) approaches \(0\) from the positive or negative side, the graph approaches the \(y\)-axis (\(x = 0\)) but never crosses it.

Answer:

E. approaches but does not cross