为什么若f(x)为偶函数,则f(x+1)=f(-x-1)? 若f(x)为偶函数则f(x+1)=f(-x+1)成立吗?
\u82e5f(x+1)\u4e3a\u5076\u51fd\u6570,\u5219f(-x+1)=f(x+1)?\u4e3a\u4ec0\u4e48\u4e0d\u662ff(-x-1)?\u4ee4g\uff08x\uff09=f\uff08x+1\uff09
\u90a3\u4e48f\uff08x+1\uff09\u662f\u5076\u51fd\u6570\uff0c\u8bf4\u660eg\uff08x\uff09\u662f\u5076\u51fd\u6570
\u800cg\uff08x\uff09\u662f\u5076\u51fd\u6570\uff0c\u8bf4\u660eg\uff08-x\uff09=g\uff08x\uff09
\u800cg\uff08-x\uff09=f\uff08-x+1\uff09
\u6240\u4ee5\u662ff\uff08-x+1\uff09=f\uff08x+1\uff09
\u81f3\u4e8ef\uff08-x-1\uff09=f\uff08x+1\uff09
\u5f53x=1\u7684\u65f6\u5019\uff0c\u5f97\u5230f\uff08-2\uff09=f\uff082\uff09
\u5f53x=3\u7684\u65f6\u5019\uff0cf\uff08-4\uff09=f\uff084\uff09
\u7b49\u7b49\u53ef\u77e5\uff0c\u5b8c\u5168\u5c31\u662ff\uff08-x\uff09=f\uff08x\uff09\u7684\u7ed3\u679c\uff0c\u8fd9\u8bf4\u660ef\uff08-x-1\uff09=f\uff08x+1\uff09\uff0c\u5f97\u5230\u7684\u662ff\uff08x\uff09\u662f\u5076\u51fd\u6570\u3002
\u4e0d\u6210\u7acb\uff0c\u82e5f(x)\u4e3a\u5076\u51fd\u6570\uff0c\u5219f(x+1)=f(-x-1)
\u5173\u952e\u5728\u4e8e\u7406\u89e3f(x)\u62ec\u53f7\u91cc\u9762\u662f\u81ea\u53d8\u91cf\uff0cf\uff08x\uff09\u662f\u5076\u51fd\u6570\uff0c\u90a3f(y)\u4e5f\u662f\uff0c\u4ee4y=x+1,f(y)=f(-y)
这里把括号中的x+a当做一个整体的自变量,所以
f(x+a)=f(-x-a)
所以f(x+1)=f(-x-1)是正确的。
f(x+1)=f(-x-1)=f[-(x+1)]
设x+1=t
则f(t)=f(-t)
所以f(x)为偶函数
偶函数的性质是f(x)=f(-x),
令x+1=t,
f(t)=f(-t)
则f(x+1)=f(-x-1),
不用对称轴来理解偶函数,那样容易错,简单代换即可
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