# scaling, power and exponential functions

In class we considered the functions
*f*(*x*) = 3^{x} and
*g*(*x*) = x^{4}, 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*.

**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:

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
*a*^{x} >
*x*^{n}
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.

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.