The user-friendly version of this content is available here.
The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.
This essential trains for: Math Kangaroo 9-10, AMC-12, AIME, AMC-10.
Functional equations are equations in which the expression of the function is unknown.
Most functional equations cannot be solved or have very complicated solutions.
Some functional equations are easy to solve by using simple changes of variable.
The easiest functional equations are the ones which, by using simple changes of variable, can be reduced to systems of equations.
Example 1: the substitution x=(1-x)
Find the functions
Given that the function is defined over the set of all real numbers, the constraint must be satisfied for any value of the variable x. In particular, we notice that the following change of variable:
turns 1-x into x:
Now, we can write the condition in two different ways:
It is easy to see that this is a system of equations with the unknowns f(x) and f(1-x). Of course, solving for f(x) gives the solution of the functional equation:
Multiply the second equation by 2 and subtract it from the first:
Solve for f(x):
Example 2: the substitution x=1/x
Another easy case is this one. Find the functions that satisfy:
First, let us notice that the maximum domain for this function does not include x=0.
with the substitution:
we get the new constraint:
which forms a linear system of two equations with two unknowns together with the initial constraint. For clarity, make the notation:
and solve the system for A and B:
Multiply the second equation by 2:
and add the first one to it:
Solve for f(x):
These are some of the simplest functional equations. More complicated problems may involve the study of certain properties of the function such as: monotony, continuity, etc.