Problem:
Explore the graphs of the functions

f(x) = 7x

and

g(x) = x7

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

Visualization:
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.