An alternative to the notation

f(x). Some very

radical but

sensible people like to write (x)f, pronounced... who knows. They never pronounce it. Maybe "x under f" would be good.

Anyhow, the reason for this is that if we're talking about

functions, we're eventually going to be wanting to

compose some of these functions.

To expand one of the examples we already have, consider the functions

(x)f = 2x + x^{2}

(x)g = (x^{2} + 1)^{-1}

The composition of the functions f and g evaluated at x is then the function

((x)f)g = ((2x + x^{2})^{2}+1)^{-1}

That is to say, first do f and then do g. Why this seemingly horrible notation is a good idea is that there are times when we need to compose a lot of functions. More than two. So suppose that we've got functions f, g, h, j, k. Their composition evaluated at x is then

(x)fghjk,

doing first f, then g, etc. (There is really no need for parentheses since function composition is associative.) In the common f(x) notation, their composition (performed in the same order) would be written

kjhgf(x),

which seems a little backwards. The last function performed is k. Nevertheless, composition on the left, f(x), is what everyone is taught; composition on the left is what I was taught, and composition on the left is what everyone does. Composition on the right might seem nifty and make sense, but if you try to do it for a while, it starts to feel like trying to write your signature by looking through a mirror. This doesn't mean it's a bad idea. It's just different.

To accomodate for us poor left-composers, some people (the same loons that came up with right composition to begin with) even modify this notation a bit further and write functions as exponents, since exponents are something we are all a bit more familiar with. Then the examples above become

x^{fg} = ((2x + x^{2})^{2}+1)^{-1}

x^{fghjk}

which starts to look more amenable, and also justifies the pronunciation "x under f".