(1)已知f(x)=2^x,g(x)是一次函数,记F(x)=f[g(x)],并且点(2,1/4)既在函数F(x)的图象上又在F^-1(x)的图象上,则F(x)的解析式为?(2)已知函数f(x)=(a^x-1)/(a^x+1)(a>1),求证f(-x)=-f(x)(3)已知x满足不等式2(log2)^2-7log2 x+3
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![(1)已知f(x)=2^x,g(x)是一次函数,记F(x)=f[g(x)],并且点(2,1/4)既在函数F(x)的图象上又在F^-1(x)的图象上,则F(x)的解析式为?(2)已知函数f(x)=(a^x-1)/(a^x+1)(a>1),求证f(-x)=-f(x)(3)已知x满足不等式2(log2)^2-7log2 x+3](/uploads/image/z/5630422-22-2.jpg?t=%281%29%E5%B7%B2%E7%9F%A5f%28x%29%3D2%5Ex%2Cg%28x%29%E6%98%AF%E4%B8%80%E6%AC%A1%E5%87%BD%E6%95%B0%2C%E8%AE%B0F%28x%29%3Df%5Bg%28x%29%5D%2C%E5%B9%B6%E4%B8%94%E7%82%B9%282%2C1%2F4%29%E6%97%A2%E5%9C%A8%E5%87%BD%E6%95%B0F%28x%29%E7%9A%84%E5%9B%BE%E8%B1%A1%E4%B8%8A%E5%8F%88%E5%9C%A8F%5E-1%28x%29%E7%9A%84%E5%9B%BE%E8%B1%A1%E4%B8%8A%2C%E5%88%99F%28x%29%E7%9A%84%E8%A7%A3%E6%9E%90%E5%BC%8F%E4%B8%BA%3F%282%29%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D%28a%5Ex-1%29%2F%28a%5Ex%2B1%29%28a%3E1%29%2C%E6%B1%82%E8%AF%81f%28-x%29%3D-f%28x%29%283%29%E5%B7%B2%E7%9F%A5x%E6%BB%A1%E8%B6%B3%E4%B8%8D%E7%AD%89%E5%BC%8F2%28log2%29%5E2-7log2+x%2B3)
(1)已知f(x)=2^x,g(x)是一次函数,记F(x)=f[g(x)],并且点(2,1/4)既在函数F(x)的图象上又在F^-1(x)的图象上,则F(x)的解析式为?(2)已知函数f(x)=(a^x-1)/(a^x+1)(a>1),求证f(-x)=-f(x)(3)已知x满足不等式2(log2)^2-7log2 x+3
(1)已知f(x)=2^x,g(x)是一次函数,记F(x)=f[g(x)],并且点(2,1/4)既在函数F(x)的图象上又在F^-1(x)的图象上,则F(x)的解析式为?
(2)已知函数f(x)=(a^x-1)/(a^x+1)(a>1),求证f(-x)=-f(x)
(3)已知x满足不等式2(log2)^2-7log2 x+3≤0,求函数f(x)=log2 x/2·log2 x/4的最大值和最小值
(1)已知f(x)=2^x,g(x)是一次函数,记F(x)=f[g(x)],并且点(2,1/4)既在函数F(x)的图象上又在F^-1(x)的图象上,则F(x)的解析式为?(2)已知函数f(x)=(a^x-1)/(a^x+1)(a>1),求证f(-x)=-f(x)(3)已知x满足不等式2(log2)^2-7log2 x+3
(1)设g(x)=kx+b,则F(x)=f[g(x)]=2^(kx+b).又F^-1(x)=1/k(log2x-b);
又点(2,1/4)既在函数F(x)的图象上又在F^-1(x)的图象上因此有:1/4=2^(2k+b)……(1),1/4=1/k(log22-b)……(2);
化简可得:2k+b=-2……(3),k+4b=4……(4);解得:k=-12/7,b=10/7.
因此,g(x)=-12/7x+10/7,F(x)= 2^(-12/7x+10/7)
(2) f(-x)=[a^(-x)-1)]/[a^(-x)+1]= (1/a^x-1)/(1/a^x+1)
=-(a^x-1)/(a^x+1)=- f(x).
(3) x满足不等式2(log2x)^2-7log2 x+3≤0,解得1/2≤log2 x≤3.
而f(x)=log2 x/2•log2 x/4=(log2 x-1)(log2 x-2)
=(log2x)^2-3log2 x+2.
f(x)= (log2x)^2-3log2 x+2是关于log2x的二次函数.其对称轴是log2x=3/2.
又1/2≤log2 x≤3,含极小值点log2x=3/2..
即f(x)最小值=(3/2)^2-3*3/2+2=-1/4.而最大值必在端点3处取得.
所以,f(x)的最大值=3^2-3*3+2=16.