Graph of Circle as a Function

Date: 05/30/98 at 10:15:32
From: Molly Thompson
Subject: circles

Is the graph of a circle a function? How could you make it a function?  
I have determined that it is not a function because when you do the 
vertical line test, the line intersects the circle in two places. But 
I don't know how you could make it a function.

Date: 05/30/98 at 13:45:04
From: Doctor Gary
Subject: Re: circles

I hope you know what the "vertical line" test tells you, and how that 
is related to the definition of a function. The "vertical line" test 
on the two-dimensional plane tells you whether there is any value of x 
for which there is more than one value of y. Since that's true for 
lots of values of x in the graph of any circle, a circle is not a 
function of x, in the same way that lines or some parabolas are.

To make the graph of a circle into a function, we'll start by looking 
at the definition of the word "function". A function is an operation 
which yields one, and only one, "result" for each item of "input." But 
there's no reason why an item of input has to be found on the x axis, 
or why the result has to be a single solitary number. The input could 
be distance along a diagonal, or the angle of a ray, and the result 
could be a point of the Cartesian plane.

Suppose, for example, that we drew a ray, beginning at the point (0,0) 
and going up and to the right, so that x and y were always equal. We 
could create a "function" which took the distance traveled "up the 
ray" as the input and the point we'd be on the line (both x and y 
coordinate) as the "result." If we travel square root of 2 up the ray, 
we are at the point (1,1). For any distance, there is only one point.   

Now let's think about making a circle into a function. If our "input" 
were the angle of a ray drawn outward from the point (0,0), our 
"result" could be the x and y coordinates of the point at which the 
ray intersected the unit circle centered at the origin. In fact, 
trigonometry is based on this very concept. For each angle, there is 
one, and only one, point at which the ray intersects the circle. The 
angles are measured counterclockwise, by reference to the right-hand 
side of the x-axis, and the co-ordinates of that point of intersection 
between the ray and the unit circle are called the cosine and sine of 
the angle.    

-Doctor Gary, The Math Forum
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