Suppose [math]F(x)[/math] is a continuous cumulative probability distribution function with [math]\lim_{x\rightarrow 1}F(x)=1[/math] and [math]F(x)\gt0[/math] for all [math]x[/math]. For which of the following [math]g(x)[/math] is [math]F(g(x))[/math] also a cumulative probability distribution function?

• [math]x^2[/math]
• [math]\sqrt{|x| + 1} [/math]
• [math]e^{-x}[/math]
• [math](1 + e^{-x})^{-1}[/math]
• [math]1-\ln(1 + e^{-x})[/math]