题目内容
已知向量



(1)求函数

(2)若集合M={f(x)|f(x)+f(x+2)=f(x+1),x∈R},试判断g(x)与集合M的关系.
【答案】分析:(1)由向量
=
,向量
与向量
关于x轴对称,及函数
,我们易求出函数的解析式,进而根据余弦函数的性质,求出函数的单调增区间;
(2)由(1)的结论,我们易对g(x)+g(x+2)进行化简,然后与g(x+1)进行比较,易得到g(x)与集合M={f(x)|f(x)+f(x+2)=f(x+1),x∈R}的关系.
解答:解:(1)∵向量
与向量
称,
∴
,
∴
=
由
,
∴g(x)区间为[6k,6k+3](k∈Z)
(2)∵
=
=
=
∴g(x)∈M
点评:本题考查的知识点是平面向量的数理积运算及演绎推理,其中根据平面向量数量积的运算计算出函数g(x)的解析式,是解答本题的关键.





(2)由(1)的结论,我们易对g(x)+g(x+2)进行化简,然后与g(x+1)进行比较,易得到g(x)与集合M={f(x)|f(x)+f(x+2)=f(x+1),x∈R}的关系.
解答:解:(1)∵向量


∴

∴

=

由

∴g(x)区间为[6k,6k+3](k∈Z)
(2)∵

=

=

=

∴g(x)∈M
点评:本题考查的知识点是平面向量的数理积运算及演绎推理,其中根据平面向量数量积的运算计算出函数g(x)的解析式,是解答本题的关键.

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