Name:  Jenna Caldwell

Title:   Zooming in on Domain and Range Principles

Description:  This lesson serves to utilize Mathematica's graphing capabilities to help students visualize the results of changing a function's domain and range in various ways.  The intended audience spans over several years, from perhaps middle school to undergraduate mathematics.  This is not a formal treatment of domain and range concepts, since they are only treated here as intervals.

Introduction

When sketching graphs of functions with chalk or pencil, human nature guides us to graph the most interesting portion of the function.  However, computers (or calculators for that matter) typically don't have the keen insight required to make such judgements; they graph exactly the portion we choose, in terms of magnitude and location of the range and x and y intervals over which to graph.  But what happens if we are'nt careful about those choices?

Motivation

f[x_]=2x
Plot[f[x],{x,-2,2}]

This is an example where the range and domain do not matter, for no matter how much we zoom in (restrict the range and domain to small centered regions of the axis, say -0.001 to 0.001), the graph looks identical.  Similarly, zooming out (enlarging the axes to say, -100 to 100) will resemble the above graph, as evident below.  
Less boring functions won't behave this way, as we'll see later.

Of course, if we change the location of the range or domain,  such as letting x be restricted to {5,10}, the graph will "float" away from the x and y axes, but it will retain its shape.

Plot[f[x],{x,5,10}]

Compare this graph to those above.  Note how a new set of axes has been used here instead of the usual x and y axes, which are relatively far away from the segment of f we chose to graph.

Now let's look at functions where domain and range choices are more critical to shape.

Examples

f(x) = Sinx Graphs

Notice how these three graphs of the same identical Sine function look quite different when we change the domain only.  The wave was "smushed" like an accordian because we didn't change the y axis spacing.  Next let's see how changing both the domain and the range can make a graph appear REALLY different, even without changing the ratio of the spacing on the axes!

f(x)=x^2 Sin(1/x) Graphs

Plot[x^2 Sin[1/x], {x, -20,20}] 
Plot[x^2 Sin[1/x], {x, -5,5}]
Plot[x^2 Sin[1/x], {x, -1,1}]
Plot[x^2 Sin[1/x], {x, -.1,.1}]

Upon activating the above cell, we see that, even when the domain and range of f are each in the  form {-a,a}, the graph of f looks quite different for various "a" values.  So, think carefully about generalizing what a graph looks like from just one sketch!

Summary

Investigate graphs by:
				 -alternate the range from {-a,a} to some other {b,c} instead
				 -smushing/stretching one of the domain, range
				 -zoom in tighter to get a closer look
				 -back up and get the "big picture"
				 -know when any of the above will not alter a graph's appearance.
This is easy to explore with quick graphing slaves!