π(π₯) = π βπ₯ 1+πβπ₯ . i) Determine whether π(π₯) is a one-to-one function.
Let's solve the equation f(x)=f(y)f(x)=f(y)f(x)=f(y) :
eβx1+eβx=eβy1+eβy\frac{e^{-x}}{1+e^{-x}}=\frac{e^{-y}}{1+e^{-y}}1+eβxeβxβ=1+eβyeβyβ
1ex+1=1ey+1\frac{1}{e^x+1}=\frac{1}{e^y+1}ex+11β=ey+11β
ex+1=ey+1e^x+1=e^y+1ex+1=ey+1
ex=eye^x=e^yex=ey
x=yx=yx=y
This means that the function f(x)f(x)f(x) is one-to-one.
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