Re: 2 pi i = 0

[Larry Smith <larry.smith@SNOW.EDU>]

At 4:45 PM -0500 11/22/02, Michael Edmiston wrote:

> Today, if I understand it correctly, all functions are relations, but
> not all relations are functions.  Today, a relation is any set of
> ordered pairs, and a function is a relation with 1-to-1 mapping.

Well, I'm old enough to remember (in fourth grade) the "black box with an
input and an output" definition of a function, but I'm also young enough to
have taught the new jargon.

Not all functions are 1-1.  The modern definitions say that a function has
to pass the vertical line test and that 1-1 functions must also pass the
horizontal line test.  Single-valued and 1-1 are not synonymous.  1-1
functions have inverses; not all single valued functions do (e.g. y = x^2
is a single-valued function but is not 1-1 and does not have an inverse).

> If we have y = sqrt(x^2+25) then y has a relation to x, but y is not a
> function of x.  The graph of y versus x is a circle centered on the
> origin with radius = 5.

The one you typed is not a circle, but _is_ a function (passes the vertical
line test).  Maybe you meant y = sqrt(25-x^2) which is also a function
because it is only the upper half-circle and therefor passes the vertical
line test (but it isn't 1-1 because it doesn't pass the horizontal line
test.  By definition the radical sign means only the positive root.

> It does not pass the 1-to-1 test, or stated
> another way the graph does not pass the "vertical line test."

They are not the same thing.

> In my school days y = sqrt(x^2 + 25) was every bit as much a function as
> y = 2x + 5.  The former was a double-valued function except single
> valued at x = +or- 5.  The latter was a single-valued function.  We did
> use the "vertical line test" but only to determine if the function was
> multi-valued or not.

I agree with the earlier comment that the current usage of "function" is
what used to be called "single-valued function."

> Bottom line, it appears to me the old-timer definition of function is
> the same as the current definition of a relation.

Sounds like so.

Larry

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.