Exponentials and Power Functions: Which Dominate? In everyday language, the word exponential is often used to imply very fast growth. But do exponential functions always grow faster than power functions? To determine what happens the “in long run,” we often want to know which functions as gets arbitrarily large. dominate x Let's consider = 2x and = The close-up view in Figure (a) shows that between = 2 and = 4, y y x3. x x the graph of = 2x lies below the graph of = The far-away view in Figure (b) shows that the y y x3. exponential function = 2x eventually overtakes = Figure (c), which gives a very far-away y y x3. view, shows that, for large the value of is insignificant compared to 2x. Indeed, 2x is growing so x, x3 much faster than that the graph of 2x appears almost vertical in comparison to the more leisurely x3 climb of x3. We say that Figure (a) a view of the functions' behavior, whereas Figure (c) gives a local global view. In fact, exponential growth function eventually dominates power function. Although an every every exponential function may be below a power function for some values of if we look at large enough x, (with > 1) will eventually dominate no matter what is. x-values, ax a xn, n
Exponentials and Power Functions: Which Dominate? In everyday language, the word exponential is often used to imply very fast growth. But do exponential functions always grow faster than power functions? To determine what happens “in the long run,” we often want to know which functions dominate as x gets arbitrarily large. Let's consider y = 2x and y = x3. The close-up view in Figure (a) shows that between x = 2 and x = 4, the graph of y = 2x lies below the graph of y = x3. The far-away view in Figure (b) shows that the exponential function y = 2x eventually overtakes y = x3. Figure (c), which gives a very far-away view, shows that, for large x, the value of x3 is insignificant compared to 2x. Indeed, 2x is growing so much faster than x3 that the graph of 2x appears almost vertical in comparison to the more leisurely climb of x3. We say that Figure (a) a local view of the functions' behavior, whereas Figure (c) gives a global view. In fact, every exponential growth function eventually dominates every power function. Although an exponential function may be below a power function for some values of x, if we look at large enough x-values, ax (with a > 1) will eventually dominate xn, no matter what n is.