Question

Functions p, q, and r are defined as:

p(t) = t4,
q(t) = 2t2,
r(t) = p(t) - 9t.
For t > 0, what describes the behavior of r as t gets farther from 0?
A) r(t) approaches +[infinity]
B) r(t) approaches -[infinity]
C) r(t) approaches +[infinity] or -[infinity] depending on the value of t
D) r(t) approaches 0

Answer 1

For t > 0, function r(t) = p(t) - 9t behaves such that it approaches +infinity as t gets farther from 0, because the t^4 term in p(t) dominates the behavior of r(t) for large values of t.

Step-by-step explanation:

Considering the functions given: p(t) = t4, q(t) = 2t2, and r(t) = p(t) - 9t, we want to explore the behavior of r as t gets farther from 0, with t > 0.

Function p(t) is a quartic function that grows rapidly as t increases. The term -9t is linear and does not grow as quickly as t4. As such, while at small values of t the linear term has some noticeable impact, as t becomes large (farther from 0), the t4 term in p(t) will dominate the behavior of r(t). In mathematical terms, the dominant term as t becomes very large is t4.

Therefore, for t > 0, as t gets farther from 0, the function r(t) will increase towards +infinity.