Basic Methods For Solving Functional Equations

  • 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 "linear" 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.

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