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1. Functional equations in one variable

1. Functional equations in one variable

1. Functional equations in one

Mathematical Database FUNCTIONAL EQUATIONS Functional equations are equations for unknown functions instead of unknown numbers. In this chapter, we will try to explore how we can find the unknown function when we know that the conditions it satisfies. 1. Functional equations in one variable Functional equations in one variable are usually easier to solve. Although there is no definite method to solve functional equations, there are some tips. Transformation of variables It is one of the most common approaches to solve a functional equation. When we apply this technique, we replace one variable by another (remember that the domain of the original variable should NOT be affected) so that a new functional equation is obtained. Sometimes it is easier for us to find the unknown function. Example 1.1. 2 If f( x+ 7) = x − 5x+ 2, find f ( x ). Solution. Let t = x+ 7 , then x = t − 7 . Directly substitution yields, 2 2 2 f( t) = ( t− 7) + 5( t− 7) + 2= t − 9t+ 16. Thus f( x) = x − 9x+ 16. Example 1.2. If 2 ⎛ x+ 1⎞ x + 1 1 f ⎜ ⎟ = + ⎝ x ⎠ x x , find f ( x ). Page 1 of 10

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