Explore the graphs of the functions

f(x) = 7x


g(x) = x7

and determine which of the two functions grows more rapidly than the other.

Use the following LiveMath Notebook to explore these graphs. The blue graph is the graph of the exponential f(x) = 7x and the red graph is the graph of the polynomial f(x) = x7. The second graph is the graph of their difference.

After a bit of exploration, It appears that, for x < 1.5301401, f(x) > g(x) and, for x > 1.5301401, that f(x) < g(x). Consider the following graph which is the graph of the quotient y = f(x)/g(x); we see that there is a dramatic change for x between 6 and 8. This suggests that we may have overlooked something. Of course, there is another intersection of the graphs of f and g at x = 7.