题目内容
已知f(x)=2x-1的反函数为f-1(x),g(x)=log4(3x+1).(1)若f-1(x)≤g(x),求x的取值范围D.
(2)设函数

【答案】分析:(1)先求出反函数的解析式及定义域,把解析式代入不等式,利用对数函数的单调性和定义域解此不等式;
(2)先利用对数的运算性质化简H(x)的解析式,再结合对数函数的图象与性质,从而解决问题.
解答:解:由y=2x-1得2x=y+1,∴x=log2(y+1)
∴f-1(x)=log2(x+1)(x>-1)
(1)由f-1(x)≤g(x)得log2(x+1)≤log4(3x+1)
∴log4(x+1)2≤log4(3x+1)
∴
∴D=[0,1]
(2)
∵0≤x≤1∴1≤x+1≤2
∴
∴
∴
∴
点评:本题考查反函数的求法和函数的值域,属于对数函数的综合题,要会求一些简单函数的反函数,掌握有关对数函数的值域的求法,属中档题.
(2)先利用对数的运算性质化简H(x)的解析式,再结合对数函数的图象与性质,从而解决问题.
解答:解:由y=2x-1得2x=y+1,∴x=log2(y+1)
∴f-1(x)=log2(x+1)(x>-1)
(1)由f-1(x)≤g(x)得log2(x+1)≤log4(3x+1)
∴log4(x+1)2≤log4(3x+1)
∴

∴D=[0,1]
(2)

∵0≤x≤1∴1≤x+1≤2
∴

∴

∴

∴

点评:本题考查反函数的求法和函数的值域,属于对数函数的综合题,要会求一些简单函数的反函数,掌握有关对数函数的值域的求法,属中档题.

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