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c 2007 The Author(s) and The IMO Compendium Group Functional Equations Marko Radovanovi´c radmarko@yahoo.com Contents 1 2 3 4 1 Basic Methods For Solving Functional Equations . Cauchy Equation and Equations of the Cauchy type Problems with Solutions . . . . . . . . . . . . . . Problems for Independent Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 2 . 2 . 14 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 1 integer. After that we find f and f (r) for rational r. This method is used in problems n where the function is defined on 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’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’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 relashionship between f ( f (n)), f (n), and n. Olympiad Training Mater...
c
2007 The Author(s) and The IMO Compendium Group
Functional Equations
Marko Radovanovi´c
radmarko@yahoo.com
Contents
1 Basic Methods For Solving Functional Equations . . . . . . . . . . . . . . . . . . . 1
2 Cauchy Equation and Equations of the Cauchy type . . . . . . . . . . . . . . . . . . 2
3 Problems with Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4 Problems for Independent Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1 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
1
n
and f (r) for rational r. This method is used in problems
where the function is deﬁned on 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’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’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 relashionship between f( f (n)), f(n),
and n.
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