Let tanh(̑) be the hyperbolic tangent function.
(a) [5 p] Show that the tanh(∙) function and the logistic sigmoid function ̑(∙) are related by:
tanh(̑) = 2̑(2̑) - 1.
[10 p] Show that a general (M-th order polynomial) linear combination of logistic sigmoid functions of the form y(x, ̑) = ̑_0 + ∑_{j=1}^M ̑_j ̑((x-̑_j)/s) is equivalent to a linear combination of tanh(∙) functions of the form y(x, u) = u_0 + ∑_{j=1}^M u_j tanh((x-̑_j)/2s), and find expressions to relate the new parameters {u_0, ..., u_M} to the original parameters {̑_0, ..., ̑_M}.