It looks as though there are many correspondences around when we consider














In particular it looks like and that .
So now let us consider what happens when How about
Now how about Can you you generalize what happens?
What numbers can m and n be? What numbers can you get when you multiply x terms? Can you multiply any two positive real numbers together to get an x value and find a y value to which it corresponds? Can every positive number be written as 2 to some power?
What other properties does the correspondence between x and y have?














Now what is the correspondence between y and x? What properties does it have?
Now lets look at a graph that you are familiar with. Click HERE to investigate the area under the curve using GSP.
Can you develop a table of values like the one above with the heights of the areas as the x values and the total area as the yvalue? Are the properties of the correspondence the same for this data as the data that we generated above?
Reflecting on patterns
How do the properties of the correspondence change when b changes?
Does each value of correspond to only one value of a?
What values of b are possible? What values of b are not possible?
What values of a are possible?