Final answer:
Exponential growth functions represent situations like bacterial colonies expanding, while exponential decay functions could model scenarios like radioactive substances decreasing over time.
Step-by-step explanation:
Exponential Growth Function
An exponential growth function has the form f(t) = a × bt, where a is the initial amount, b is the base or growth factor (when b > 1), and t is time.
For example, consider a bacteria population that doubles every hour, starting with one bacterium.
The function would be f(t) = 1 × 2t. After 5 hours, this would equal f(5) = 32, which shows the population grew to 32 bacteria.
A scenario that represents this exponential growth is the proliferation of a bacteria colony in a petri dish with unlimited nutrients. Initially, the population grows slowly, but as bacteria multiply, the growth rate increases dramatically, forming a J-shaped curve
Exponential Decay Function
Conversely, an exponential decay function is typically written as g(t) = a × b-t, where a is the initial amount, t is time, and b is the decay factor (when 0 < b < 1).
An example is the radioactive decay of a substance, where half of the material decays every fixed period. If we have 100 grams of a substance that halves every year, the decay function is g(t) = 100 × (1/2)t.
In a real-world situation, this could represent the amount of a particular pharmaceutical drug in the bloodstream, where it halves every certain number of hours due to metabolism, showcasing an exponential decrease over time.