Question

求函数f(X)=f(x)=2^x/2^x+2^-x的反函数f^-1(x)的解析式和定义域。详解 谢谢

Answer 1

f(x)=2^x/2^x+2^-x
=2^x/[2^x+1/2^x]
=4^x/(4^x+1)
=1-1/(4^x+1)
值域为(0,1)

y=4^x/(4^x+1)
y4^x+y=4^x
(y-1)4^x=-y
4^x=-y/(y-1)
x=(1/2)log2[y/(1-y)]
反函数为
y=(1/2)log2[x/(1-x)]
定义域为(0,1)

追问

我想问下这里=4^x/(4^x+1)
=1-1/(4^x+1)中4^x怎么化成1-1？

追答

=4^x/(4^x+1)
=(4^x+1-1)/(4^x+1)
=1-[1/(4^x+1)]
有个括号

追问

4^x=-y/(y-1)中-y/(y-1)怎么化为y/1-y

追答

-y/(y-1)把负号移到括号里面
y/(1-y)