抽象函数是指那些没有给出解析式的函数,因为缺少具体的表达式,所以分析和解决这类问题时感到棘手,如果能根据条件的特征,采用变量代换法,创造从难到易转化的条件,那么问题往往得以圆满地解答. 例1 已知函数f(x)对任意x1,x2∈R,都有f(x1)+f(x2)=2f(x1+x2/2)· 不恒为零. 求证:(1)f(x)是偶函数; (2)f(x)是周期为2π的周期函数. 证明(1)不妨设f(x0)≠0,取x1=x2=x0,得2f(x0)=2f(x0)f(0),则f(0)=1. 又取x1=x,x2=-x(x∈R),得 Abstract functions are those functions that do not give analytical expressions. Because they lack concrete expressions, they are intractable when analyzing and solving such problems. If they can be based on the characteristics of conditions, variable substitution method is used to create difficult to easy transformations. The conditions, then the problem is often successfully answered. Example 1 Known function f (x) for any x1, x2 ∈ R, have f (x1) + f (x2) = 2f (x1 + x2/2) · not Constant zero. Proof: (1) f(x) is an even function; (2) f(x) is a periodic function with a period of 2π. Proof (1) Let f(x0) ≠ 0 be taken and x1 = x2 = X0, get 2f (x0) = 2f (x0) f (0), then f (0) = 1. Also take x1 = x, x2 = -x (x ∈ R), get