(purple)

(red)

(blue)

(green)

By studying the graph, there appears to be two different shapes: a graph that is bound within x=[-1,1] and y=[-1,1] and a graph that has two tails. Below I have divided up the above graph into two graphs: the first showing even exponents and the second showing odd exponents.

Even exponents: 2 and 4

Odd exponents: 3 and 5

Now two questions remain: first, why does the function differ depending on whether the exponents are even or odd and second, what will the graphs of larger exponents like 24 and 25 look like? Will they follow the same pattern we see here?

Let's explore the first question to determine the difference between even and odd exponents. Mathematically, lets look at the values of x and y in the range (-1.5,-.6) by setting each equation equal to y.

x =	y = (1-x^2)^(1/2)	y = (1-x^3)^(1/3)	y = (1-x^4)^(1/4)	y = (1-x^5)^(1/5)
-1.5	undefined	1.6355	undefined	1.5376
-1.4	undefined	1.5528	undefined	1.4486
-1.3	undefined	1.4732	undefined	1.3635
-1.2	undefined	1.3973	undefined	1.2839
-1.1	undefined	1.3259	undefined	1.2116
-1.0	0	1.2599	0	1.1487
-.9	.43589	1.2002	.76579	1.0973
-.8	.6	1.1478	.87657	1.0583
-.7	.71414	1.1033	.93366	1.0316
-.6	.8	1.0674	.96589	1.0151

The functions above represent only the positive portion of each function, therefore we are only checking a few negatives values where the all the functions differ the most. As illistrated in the table, the two functions above that are undefined when x is less than -1 are even exponent functions. On the other hand, the odd exponent functions are defined at x values less than -1. For a closer look at why this happens, lets look at the actual equation beginning with even functions under the radicals. In mathematics, there cannot be a negative under an even valued radical because it is impossible to multiply a negative number by itself an even number of times to get a negative number. Therefroe for the the even exponent functions 1 - x^2 must be positive. Through the exploration, where these functions become undefined is the point where there is a negative under the radical. For example, if x = -2 then 1 - (-2)^2 = -3 and therefore there does not exist a square root for -3, hence it is undefined for all even exponent functions. Yet for odd valued exponents, there can exist both a positive and negative value under the radical because a negative number multiplied by itself an odd number of times is a negative number. For example, if x = -2 again then 1 - (-2)^3 = -7 which has a rational cube root.

The negative portion of the functions under the x-axis react the same way as the postive portion because the positive portion is symmetrical to the bottom portion except the values that are undefined for even exponent functions are x > 1 instead of x < -1.

Understanding these concepts, what would and look like? Well the exponent 24 is going to make the function undefined from x < -1 and x > 1 while the exponent 25 will be defined through the real numbers. Below is a graph of the two functions:

(red)

(purple)