Consider an exponential function
with base 𝑎. For which values of 𝑎 is the
In this problem, we are given an
exponential function 𝑓 of 𝑥 equals 𝑎 to the 𝑥. Recall that the base 𝑎 may be
given by 𝑓 of 𝑥 over 𝑓 of 𝑥 minus one. Recall that 𝑎 is a constant real
number with 𝑎 greater than zero and not equal to one. This means that regardless of the
value of 𝑥, the value of 𝑓 of 𝑥 over 𝑓 of 𝑥 minus one is always a constant. Therefore, for any value of 𝑥, the
increase in 𝑓 of 𝑥 between 𝑥 minus one and 𝑥 is always 𝑎.
Since an exponential function is
always positive, 𝑓 of 𝑥 over 𝑓 of 𝑥 minus one being greater than one implies
that 𝑓 of 𝑥 is greater than 𝑓 of 𝑥 minus one and, therefore, that the function
is increasing. And likewise, 𝑓 of 𝑥 over 𝑓 of
𝑥 minus one being less than one implies that 𝑓 of 𝑥 is less than 𝑓 of 𝑥 minus
one and, therefore, that the function is always decreasing.
Since the left-hand side of both of
these inequalities is equal to 𝑎, this means that 𝑎 greater than one implies the
function increases and 𝑎 less than one implies it decreases. Remember also that 𝑎 must be
greater than zero, so this gives us our final answer. 𝑎 greater than one implies the
function increases and 𝑎 between zero and one implies the function decreases.