Substituting the values for variables.
The most common first attempt is with some constants
(eg. 0 or 1), after that (if possible)
some expressions which will make some part of the equation
to
become constant. For example if \(f(x+y)\) appears in
the equations and if we have found \(f(0)\) then we plug
\(y=-x\). Substitutions become less obvious as the
difficulty of the problems increase.
Mathematical induction. This method relies
on using the value \(f(1)\) to find all
\(f(n)\) for \(n\) integer. After that we find
\(f\Big(\frac 1n\Big)\)
and \(f(r)\) for rational \(r\). This method is used in
problems where the function is defined on
\(\mathbb{Q}\) and is very useful, especially with
easier problems.
Investigating for injectivity or
surjectivity of
functions involved in the equaiton.
In many of the problems these facts are not difficult to
establish but can be of great importance.
Finding the fixed points or zeroes of functions.
The number of problems using this method is considerably
smaller than the number of problems using some of the previous
three methods. This method is mostly encountered in more difficult
problems.
Using the Cauchy\(\prime\)s equation
and equation of its type.
Investigating the monotonicity and continuity of
a function. Continuity is usually given as additional condition
and as the monotonicity it usually serves for reducing the problem
to Cauchy\(\prime\)s equation. If this is not the case, the
problem is on the other side of difficulty line.
Assuming that the function at some point is
greater or smaller then the value of the function for which
we want to prove that is the solution. Most often
it is used as continuation
of the method of mathematical induction
and
in the
problems in which the range is bounded from either side.
Making recurrent relations.
This method is usually used with the equations in which
the range is bounded and in the case when we are able
to find a relationship between \(f(f(n))\), \(f(n)\), and
\(n\).
Analyzing the set of values for which
the function is equal to the assumed solution. The goal
is to prove that the described set is precisely the
domain of the function.
Substituting the function.
This method is often used
to simplify the given equation and is seldom of crucial
importance.
Expressing functions as sums of odd and even.
Namely each function can be represented as a sum of one
even and one odd function and this can be very
handy in treating _quot_linear_quot_ functional equations involving
many functions.
Treating numbers in a system with basis different than
\(10\). Of course, this can be used only if the domain is
\(\mathbb {N}\).
For the end let us emphasize that it is very
important to guess the solution at the beginning. This
can help a lot in finding the appropriate substitutions.
Also, at the end of the solution, DON\(\prime\)T FORGET to verify
that your solution satisfies the given condition.