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Functional Equation

Date: 02/18/2008 at 00:46:51
From: Jack
Subject: solve for f(x)

If 2f(x) + f(1-x) = x^2 for all x, then f(x) = ?

First I tried to solve for x, and get x alone.  Once I got x alone, I 
went further but just ended up confusing myself.  Any help would be 
appreciated.



Date: 03/16/2008 at 21:43:31
From: Doctor Ali
Subject: Re: solve for f(x)

Hi, Jack!

Thanks for writing to Dr. Math.  Let's start with what you said:

  2f(x) + f(1-x) = x^2

This equality is TRUE for all x's.  So, we can change x to anything 
and the equality remains TRUE.  For example, by what you said, we can 
deduce that whatever f(x) may be, the following equalities are TRUE:

Substitute zero for x.

  1) 2f(0) + f(1) = 0

Substitute one for x.

  2) 2f(1) + f(0) = 1

Note that 1 and 2 are both expressions with respect to f(0) and f(1)
and both of them are TRUE.  We can therefore see them as a system of
simultaneous equations in terms of f(0) and f(1).  Letís continue:

Assume that f(0) = A and f(1) = B and letís re-write the expressions:

  1) 2A + B = 0

  2) 2B + A = 1

Solve the equations for A and B and youíll get:
 
  A = -1/3

  B = 2/3

Now we know that whatever f(x) may be, f(0) has to be -1/3 and f(1) 
has to be 2/3.

So, what's the use?  We know only two special values of x.  But letís 
generalize the method.

As you may have noticed, when we substituted 1 for x, we got an 
expression with respect to f(0) and f(1), the two values we got in 
the first substitution and not f(2) or something else that was not in
the first one.  Having both expressions involve f(0) and f(1) allowed
us to go on to the step of solving the system of equations.  So that's
the key idea.

We said that we can change x into anything.  Here ďanythingĒ can be 
some kind of expression itself.  For example, it can be ď2xĒ, ď1/xĒ 
or whatever.  Among all of these expressions, we are looking for the 
one that, when simplified, gives us a system of equations and 
doesnít make new variables.  By solving the system, we can find the 
answer.

Letís try your equation again:

  2f(x) + f(1-x) = x^2

What should I change x into in order to get the system?  Remember that
we want to wind up with f(x) and f(1-x) again after our substitution
so that we will have a system of equations with the new equation and
the original one.

Is it ď2xĒ?  Letís try:

  2f(2x) + f(1-2x) = 4x^2

Wow!  Now I have two other unknowns with f(2x) and f(1-2x).  So 2x
didnít work.  What about ďx+1Ē?

  2f(x+1) + f(1-(x+1)) = (x+1)^2

Or
 
  2f(x+1) + f(-x) = x^2 + 2x + 1

We get f(-x) and f(x+1) but those aren't in the original equation. 
Letís try ď1-xĒ:

  2f(1-x) + f(1-(1-x))=(1-x)^2

Or

  2f(1-x) + f(x) = x^2 - 2x + 1

It worked.  Now we have f(x) and f(1-x), as in the original 
expression.  Whatís good about this?  Remember the system of 
simultaneous equations.

  1) 2f(x) + f(1-x) = x^2

  2) 2f(1-x) + f(x) = x^2 - 2x + 1

In order not to confuse ourselves, letís assume that A = f(x) and 
B = f(1-x).

  1) 2A + B = x^2

  2) 2B + A = x^2 - 2x + 1

Solve the system to get A, which is f(x).

Note:  This kind of equation is called a "FUNCTIONAL EQUATION", which
means that you are not trying to find x's that satisfy the equality
(as we do in ordinary equations), but instead are trying to find 
FUNCTIONS like f(x) that satisfy the equality.

Please write back if you still have any difficulties.

Bye!

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Basic Algebra
High School Functions

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