Functions

Let f₁ : R \(\to\) R, f₂ : [0, \(\infty\)) \(\to\) R, f₃ : R \(\to\) R, and f₄ : R \(\to\) [0, \(\infty\)) be defined by

\({f_1}\left( x \right) = \left\{ {\matrix{ {\left| x \right|} & {if\,x < 0,} \cr {{e^x}} & {if\,x \ge 0;} \cr } } \right.\)

f₂(x) = x² ;

\({f_3}\left( x \right) = \left\{ {\matrix{ {\sin x} & {if\,x < 0,} \cr x & {if\,x \ge 0;} \cr } } \right.\)

and

\({f_4}\left( x \right) = \left\{ {\matrix{ {{f_2}\left( {{f_1}\left( x \right)} \right)} & {if\,x < 0,} \cr {{f_2}\left( {{f_1}\left( x \right)} \right) - 1} & {if\,x \ge 0;} \cr } } \right.\)