scaling, power and exponential functions

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In class we considered the functions f(x) = 3x and g(x) = x4, and looked at them on different domains to see how the graphs appeared. This web explanation looks at this again.

Suppose we looked at the two functions, and picked a really big range of x and y values to consider. Then we might see something that looks like the figure shown below. Notice that because the vertical scale is huge we can't make out the difference in values between f(x) and g(x) for small values of x.

image of f(x) and g(x) for 0<=x<10 and 0<y<14,000
figure 1: the graph of f(x) and g(x)

Now suppose that we restricted our view to a smaller range of values. You can do this by selecting the x and y range you want from the selections below:

  reset to original figure
  0 <= x <= 10, 0 <= y <= 10
  0 <= x <= 6, 0 <= y <= 200
  0 <= x <= 10, 0 <= y <= 15,000

Notice as you look at each figure how the relative appearance of the graphs of f(x) and g(x) change, and make sure that you can see how the small graph is the portion of the original that is marked on the original graph.

You can do this with your graphing calculator, too. Check to see that you get the same thing as is shown here when graph these two functions and adjust the window you look at. (Note that I've estimated the ranges slightly, and I obviously didn't use a graphing calculator to generate these, so you might get slightly different figures.)

One last point: also notice that growing exponentials dominate power functions. For large enough values of x, the exponential function will always be bigger than the power function. This is true no matter which exponential we choose, and no matter which power function. This is ax > xn for any a (where a >1) and any n, as long as we consider large enough values of x. However, notice that for smaller values of x this may not be the case.
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scaling, power and exponential functions
Last modified: Thu Jan 20 16:42:44 EST 2005
Comments to:glarose(at)umich(dot)edu
©2005 Gavin LaRose, UM Math Dept.