Question Video: Discussing the Monotonicity of Exponential Functions Mathematics

Consider an exponential function with base 𝑎. For which values of 𝑎 is the function decreasing?

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Video Transcript

Consider an exponential function with base 𝑎. For which values of 𝑎 is the function decreasing?

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.

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