1. Functional equations in one variable

Method of undetermined coefficients
Mathematical Database
When we know that the unknown function satisfies certain conditions, say it is a quadratic or
2
cubic function, we can immediately set up variables (e.g. let f ( x)
= ax + bx+ c if f ( x ) is a
quadratic polynomial) and solve for them.
Example 1.6.
If f ( x)
is a quadratic function such that f ( x+ 1) − f( x) = 8x+ 3 and f (0) = 5 , find f ( x ).
Solution.
Let
2
f ( x)
ax bx c
Simplifying gives
= + + , then ax ( + 1) 2 + bx ( + 1) + c−ax 2 −bx− c= 8x+3.
After solving, we have a = 4 and b =−1.
Putting
x = 0, we have c = 5 .
2ax + a + b = 8x
+ 3.
Therefore
2
f ( x) = 4x − x+ 5.
2. Functional equations in more than one variable
For functional equations with more than one variable, we can also apply the methods
mentioned above. Besides we can also try to substitute some special values, say x = y = 0 into the
given condition given to obtain some results. As it is very difficult to describe these techniques in
words, we will try to see their application in various examples below.
Example 2.1.
If f : →
satisfies
1. f (1) = 2 ,
2. For all xy∈ , , f( xy) = f( x) f( y) − f( x+ y) + 1, find f ( x ).
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