Why use f(x)?

Date: 02/14/99 at 22:27:02
From: ed brown
Subject: Why use f(x)?

Can you explain the point of using f(x) in an equation as opposed to y?

Thanks in advance.

Date: 02/15/99 at 08:42:50
From: Doctor Jerry
Subject: Re: Why use f(x)?

One can use the notation y = x^2-5x+6 for many purposes. The equation 
can carry the idea that as x changes, y changes. It doesn't give, 
explicitly, the name of the rule that says that to the number x 
corresponds the number x^2-5x+6. 

When we write y = sin(x), we are being redundant, for some purposes. 
"sin" is the name of a particular rule. Out of habit, we may say "graph 
the equation y = sin(x)." We could say: graph the function sin.

I expect that you also use log or ln, tan, cos, and maybe sqrt(x).

Mathematicians often like to be minimalists. If they have a certain 
rule of correspondence in mind, they may prefer naming it by a letter, 
such as f. They don't need y = f(x); just f will do.

When we want to put two functions together, like ln(sin(x)), we can 
name the new function as ln o sin. Of course, this can also be done, as 
you may prefer, by saying something like

Y = ln(w)

w = sin(x),

so, y = ln(sin(x)).

Another reason is that calculus is the study of functions, not 
equations. We study functions f and their derivatives f'. We integrate 
functions. All of this is possible with the y = x^2+5x+6 notation, but 
somewhat more awkward, and it takes more talking to be precise.

Finally, if f(x) = x^2 and we want to calculate (as is needed in 
calculus) the difference quotient (this is the slope of the line 
joining (x,f(x)) and (x+h,f(x+h)))

[f(x+h)-f(x)]/h,

how could you easily give the recipe using the y = x^2 notation?  

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/