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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

equation

that satisfy:

equation

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:

equation

turns 1-x into x:

equation

Now, we can write the condition in two different ways:

equation

equation

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:

equation

equation

Multiply the second equation by 2 and subtract it from the first:

equation

equation

Solve for f(x):

equation

equation

Example 2: the substitution x=1/x

Another easy case is this one. Find the functions that satisfy:

equation

First, let us notice that the maximum domain for this function does not include x=0.

with the substitution:

equation

we get the new constraint:

equation

which forms a linear system of two equations with two unknowns together with the initial constraint. For clarity, make the notation:

equation

equation

and solve the system for A and B:

equation

equation

Multiply the second equation by 2:

equation

and add the first one to it:

equation

Solve for f(x):

equation

equation

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.