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2001年高考数学(理科)第22题如下: 定义在R上的偶函数f(x)的图象关于x=1对称,且f(1)=a>0,若对任意x1,x2∈[0,1/2]都有f(x1十x2)=f(x1)·f(x2).(1)求f(1/2)及f(1/4);(2)证明:f(x)是周期函数;(3)即an=f(2n-1/2n),求limn→∞(lian). 本题主要考察函数的概念、图象,函数的奇偶性和周期性及数列的极限等基础知识;考察运算能力和逻辑思维能力. 有关函数f(x)的问题源于教材的例、习题,但为数不多,虽能引起学生的注意,却很难形成解题技巧.为此本文通过几个例子阐述处理f(x)问题的常见方法.
The mathematics (science) question 22 of the college entrance examination in 2001 is as follows: The image of the even function f(x) defined on R is symmetric about x=1, and f(1)=a>0, if any x1, x2∈[ 0, 1/2] has f(x1 x x2) = f(x1) f(x2). (1) find f(1/2) and f(1/4); (2) prove: f( x) is a periodic function; (3) that is an = f (2n-1/2n), find limn → ∞ (lian). This question mainly examines the concept of the function, the image, the parity and periodicity of the function, and the limit of the series. Fundamental knowledge, such as ability to study and logical thinking. The problem of function f(x) stems from examples of textbooks and exercises. However, it is very limited. Although it can arouse the attention of students, it is difficult to develop problem solving skills. This article illustrates a common way of dealing with f(x) problems with several examples.