Final answer:
Both exponential functions f(x) and g(x) have the same y-intercept at (0,1), but g(x) increases faster than f(x) because it has a larger base (1.7 compared to 1.2). The exponential relationship model shows that as x increases, the growth rate of the function also increases—a concept seen in real-world examples like bacterial population growth.
Step-by-step explanation:
Comparing Exponential Functions f(x) and g(x)
The functions given, f(x) = (1.2)^x and g(x) = (1.7)^x, are both exponential functions, which show the characteristic of an exponential relationship where a change in the independent variable x results in a proportional change in the dependent variable. We can compare the rates of increase by looking at their bases. Since the base of g(x) is greater than that of f(x) (1.7 > 1.2), g(x) increases faster than f(x) as x gets larger. The y-intercepts for both functions occur when x=0, resulting in f(0) = (1.2)^0 = 1 and g(0) = (1.7)^0 = 1. Therefore, both functions have the same y-intercept at (0,1).
Exponential growth (such as the growth of bacteria under ideal conditions) is a prime example of the behavior of these functions. With each succeeding generation, the quantity of bacteria increases, accelerating the overall growth rate of the population. This concept is directly comparable to our functions, where the growth rate speeds up as the value of x increases, due to the exponential nature of the relationship.
The growth rates of f(x) and g(x) can be visualized on a graph, which is often a more common tool for such comparisons than algebraic expressions. Still, understanding the mathematics behind the graphs is crucial for a deeper understanding of growth rates and other related concepts, such as the time it takes for an investment to grow or the increase in size of a population over time.