设函数f(x)=(a·2的x次方-1)/1+2的x次方(a∈R),均满足f(-x)=-f(x).
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f(x)=(a*2^x-1)/(1+2^x)
f(-x)=(a*2^(-x)-1)/悄穗(1+2^(-x))=(a-2^x)/(2^x+1)
f(x)是R上的奇函数,f(x)=-f(-x)
(a*2^x-1)/(1+2^x)=-(a-2^x)/(2^x+1)
a*2^x-1=-a+2^x
(a-1)*(2^x+1)=0
a=1
f(x)=(2^x-1)/(2^x+1)
2)
f(x)=(2^x-1)/(2^x+1)=1-2/(2^x+1)<1
2^x+1>1
-2/(2^x+1)>-2
f(x)=1-2/(2^x+1)>1-2=-1
-1<f(x)<1
函数启兄卜f(x)的值域(-1,1)
3)
设x1<x2
f(x1)-f(x2)=[1-2/(2^x1+1)]-[1-2/(2^x2+1)]
=2/(2^x2+1)-2/(2^x1+1)
=2(2^x1-2^x2)/(2^x2+1)(2^x1+1)
<0
f(x)在R上的单调增
0<f(2x-1)<15/17
即f(0)<f(2x-1)<f(4)
即0<2x-1<4
得:1/尘差2<x<5/2
f(-x)=(a*2^(-x)-1)/悄穗(1+2^(-x))=(a-2^x)/(2^x+1)
f(x)是R上的奇函数,f(x)=-f(-x)
(a*2^x-1)/(1+2^x)=-(a-2^x)/(2^x+1)
a*2^x-1=-a+2^x
(a-1)*(2^x+1)=0
a=1
f(x)=(2^x-1)/(2^x+1)
2)
f(x)=(2^x-1)/(2^x+1)=1-2/(2^x+1)<1
2^x+1>1
-2/(2^x+1)>-2
f(x)=1-2/(2^x+1)>1-2=-1
-1<f(x)<1
函数启兄卜f(x)的值域(-1,1)
3)
设x1<x2
f(x1)-f(x2)=[1-2/(2^x1+1)]-[1-2/(2^x2+1)]
=2/(2^x2+1)-2/(2^x1+1)
=2(2^x1-2^x2)/(2^x2+1)(2^x1+1)
<0
f(x)在R上的单调增
0<f(2x-1)<15/17
即f(0)<f(2x-1)<f(4)
即0<2x-1<4
得:1/尘差2<x<5/2
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