Solving Functional Equations
Solving functional equations can be very difficult but there are some common methods of solving them. For example, in dynamic programming a variety of successive approximation methods are used to solve Bellman's functional equation, including methods based on fixed point iterations. The main method of solving elementary functional equations is substitution. It is often useful to prove surjectivity or injectivity and prove oddness or evenness, if possible. It is also useful to guess possible solutions. Induction is a useful technique to use when the function is only defined for rational or integer values.
A discussion of involutary functions is useful. For example, consider the function
Composing f with itself gives
Many other functions also satisfy the functional equation :, including
Example 1: Find all functions f that satisfy
for all assuming ƒ is a real-valued function.
Let x = y = 0
So ƒ(0)2 = 0 and ƒ(0) = 0.
Now, let y = −x:
A square of a real number is nonnegative, and a sum of nonnegative numbers is zero iff both numbers are 0. So ƒ(x)2 = 0 for all x and ƒ(x) = 0 is the only solution.
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