f(x)的定義域顯然為x∈R
設(shè)x1,x2∈R,且x1>x2
則f(x1)-f(x2)
=(2^x1 -1)/(2^x1 +1) - (2^x2 -1)/(2^x2 +1)
={[(2^x1 -1)*(2^x2 +1)] - [(2^x2 -1)*(2^x1 +1)]} / [(2^x1 +1)*(2^x2 +1)]
=2*(2^x1 - 2^x2) / [(2^x1 +1)*(2^x2 +1)] ①
函數(shù)y=2^x是基本對(duì)數(shù)函數(shù)之一,很容易知其為定義在R上的單調(diào)增函數(shù),即,當(dāng)x1>x2時(shí),有2^x1 > 2^x2 ; 且,對(duì)于任意的x,都有2^x >0,故2^x +1>0
由此可知,①式中,分子分母的各個(gè)公因項(xiàng)均大于0,故f(x1)-f(x2)>0
f(x1)>f(x2)
∴f(x)為定義在R上的增函數(shù)