函数f(x)的定义域为D=﹛x/x≠0﹜且满足对于任意x1,x2∈0,有f(x1乘x2)=f(x1)+f(x2)

问题描述:

函数f(x)的定义域为D=﹛x/x≠0﹜且满足对于任意x1,x2∈0,有f(x1乘x2)=f(x1)+f(x2)
⑴.求f(1)值
⑵判断f(x)的奇偶性
⑶如果f(4)=1,f(3x+1)+f(2x-6)≤3且f(x)在(0,+∞)上是增函数,求x的取值范围
1个回答 分类:数学 2014-11-02

问题解答:

我来补答
1) 令x1 = x2 = 1;带入f(x1x2)=f(x1)+f(x2);
f(1) = f(1) + f(1) ===> f(1) = 0;
2) 令x1 = x2 = -1;带入f(x1x2)=f(x1)+f(x2);
0 = f(1) = f(-1) + f(-1) ===> f(-1) = 0;
令x1 = -1;带入f(x1x2)=f(x1)+f(x2);
f(-x2) = f(-1) + f(x2) ===> f(-x2) = f(x2);===> f(x)是偶函数
3) f(4*4) = f(4) + f(4) = 2
f(4*4*4) = f(4*4)+f(4)=2+1=3
所以 f(64) = 3 ,f(x)是偶函数,所以,f(-64) = 3
f(3x+1)+f(2x-6) = f( (3x + 1)(2x - 6))
 
 
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