Question

𝑓(𝑥) = 𝑒 −𝑥 1+𝑒−𝑥 . i) Determine whether 𝑓(𝑥) is a one-to-one function. 

Answer 1

Let's solve the equation $f(x)=f(y)$ :

$\frac{e^{-x}}{1+e^{-x}}=\frac{e^{-y}}{1+e^{-y}}$

$\frac{1}{e^x+1}=\frac{1}{e^y+1}$

$e^x+1=e^y+1$

$e^x=e^y$

$x=y$

This means that the function $f(x)$ is one-to-one.