General Strategy for Questions on Functions

Date: 06/10/2004 at 09:04:57
From: Renu 
Subject: ACT question on functions

If the function f satisfies the equation f(x + y) = f(x) + f(y) for
every pair of real numbers x and y, what is (are) the possible 
value(s) of f(0)?

A. any real number
B. any positive real number
C. 0 and 1 only
D. 1 only
E. 0 only

Thanks a google!

Date: 06/10/2004 at 10:15:46
From: Doctor Peterson
Subject: Re: ACT question on functions

Hi, Renu.

There are two things I can see to do to start analyzing this 
functional equation to see what it says about f(0).  One is to let x 
or y be 0 and see what happens; the other is to let x + y be zero. 
First, if y = 0, we know that

  f(x + 0) = f(x) + f(0)  for any x

Since x + 0 is x, this means

  f(x) = f(x) + f(0)  for any x

What does that tell you about the value of f(0)?

Setting x + y to 0 (that is, taking y = -x) will give a different 
fact, not relevant to the question that was asked, but still 
interesting.

This problem illustrates a useful thing to keep in mind; when you 
don't already know techniques for solving a problem (and functional 
equations don't have a lot of standard techniques!), you can just 
play with what you've got and see what happens.  Try plugging things 
in and look for possibly useful implications.  In other words, even 
when you're taking a test, you can learn just the way an infant does, 
putting things into his mouth and pushing them around to see if they 
do anything worth observing.  Mathematicians are just like little 
kids - or rather, little kids are mathematicians in disguise!

If you have any further questions, feel free to write back.

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