Question

If f(x) be defined on [−2, 2] and is given by \[f\left( x \right) = \begin{cases}- 1, & - 2 \leq x \leq 0 \\ x - 1, & 0 < x \leq 2\end{cases}\]  and g(x)

\[= f\left( \left| x \right| \right) + \left| f\left( x \right) \right|\] , find g(x).

 

 

 

Answer 1

Given:

\[f\left( x \right) = \begin{cases}- 1, & - 2 \leqslant x \leqslant 0 \\ x - 1, & 0 < x \leqslant 2\end{cases}\]

Thus,

\[g\left( x \right) = f\left( \left| x \right| \right) + \left| f\left( x \right) \right|\]

\[= \begin{cases}x - 1 + 1 , & - 2 \leqslant x \leqslant 0 \\ x - 1 + ( - x + 1), & 0 < x < 1 \\ x - 1 + x - 1, & 1 \leq x \leq 2\end{cases}\]

\[ = \begin{cases}x, & - 2 \leqslant x \leqslant 0 \\ 0, & 0 < x < 1 \\ 2x - 2, & 1 \leq x \leq 2\end{cases}\]