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### 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

```

```
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/
```
Associated Topics:
High School Functions

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