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1. Functional equations in one variable

1. Functional equations in one variable

Method of

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 ). Page 4 of 10

Solution. Putting y = 1, then Mathematical Database f( x) = f( x) f(1) − f( x+ 1) + 1 = 2 f( x) − f( x+ 1) + 1 f( x+ 1) = f( x) + 1 Therefore applying condition 1 and by mathematical induction, for all integer x, we have f ( x) = x+ 1. m m For any rational number, let x = where m, n are integers and n is not zero. Putting x = , y = n, n n ⎛ then ( ) m ⎞ ⎛m ⎞ f m = f ⎜ ⎟( n+ 1) − f ⎜ + n⎟ 1 ⎝ n ⎠ ⎝ n ⎠ + . Since f( x+ 1) = f( x) + 1 for ∀x ∈ , we have ⎛m ⎞ ⎛m⎞ f ⎜ + n⎟= f ⎜ ⎟+n. ⎝ n ⎠ ⎝ n ⎠ ⎛m⎞ ⎛m⎞ Substituting this into the original equation, we have m+ 1 = f ⎜ ⎟( n+ 1) − f ⎜ ⎟−n+ 1. Thus ⎝ n ⎠ ⎝ n ⎠ ⎛m⎞ m f ⎜ ⎟ = + 1. ⎝ n ⎠ n So we have f ( x) = x+ 1 ∀x ∈ . Remark. In this question we can see that we first solve the functional equation for a special case (we found f ( x ) when x ∈ ), then we solve for the more general case. (we found f ( x ) when x ∈ ). Indeed, this method is used in many occasions. For find when f ( x ) when x∈ about this later. x ∈ . Then, find f ( x ) when x ∈ by substituting f : → , we can apply this method similarly. First, m x = . Finally, find f ( x ) n by the density of rational numbers (for continuous functions only). We shall see more Example 2.2. If 2 2 ( x y) f( x y) ( x y) f( x y) 4 xy( x y ) − + − + − = − for all x, y , find f ( x ). Solution. The given condition is equivalent to Page 5 of 10

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