Given the sigmoid function:
$$s(x) = \frac{1}{1 + e^{-x}}$$
And the $\tanh$ function:
$$\tanh(x) = 2s(2x) - 1$$
It makes sense to me that we would:
1) scale the y-axis to "stretch" the graph's range to $(0,2)$ from $(0,1)$
2) subtract 1 to "move" the graph down to the range $(-1,1)$
Please explain why we scale the $x$ ie $s(2x)$? An intuitive explanation would be valuable to understand what is happening here. Is this just by definition?
By definition we have
$$\tanh x := \dfrac{\sinh x}{\cosh x} = \dfrac{e^x-e^{-x}}{e^x+e^{-x}}=\dfrac{e^x+e^{x}-e^{x}-e^{-x}}{e^x+e^{-x}}=2\dfrac{e^x}{e^x+e^{-x}}-1$$
$$=2\dfrac{e^xe^{-x}}{e^xe^{-x}+e^{-x}e^{-x}}-1=2\dfrac{1}{1+e^{-2x}}-1=2s(2x)-1$$