Limits Continuity And Differentiability

Let \(g:R \to R\) be a differentiable function with \(g(0) = 0\), \(g'(0) = 0\) and \(g'(1) \ne 0\). Let

\(f(x) = \left\{ {\matrix{ {{x \over {|x|}}g(x),} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.\)

and \(h(x) = {e^{|x|}}\) for all \(x \in R\). Let \((f\, \circ \,h)(x)\) denote \(f(h(x))\) and \((h\, \circ \,f)(x)\) denote \(f(f(x))\). Then which of the following is (are) true?