Discussion

Explanation:

F(x) = $\left\{\begin{array}{lll}1-x& wnen& 0\le x<2\\ 1& when& -2<x<0\end{array}\right.$ and F1(x) = $\left\{\begin{array}{lll}-1-x& when& -2<x<0\\ -1& when& 0\le x<2\end{array}\right.$

Therefore, replacing x by (−x) in above functions, we get

F(-x) = $\left\{\begin{array}{lll}1+x& when& -2<x<0\\ 1& when& 0\le x<2\end{array}\right.$ and F1(-x) = $\left\{\begin{array}{lll}-1+x& when& 0\le x<2\\ -1& when& -2<x<0\end{array}\right.$

∴ -F(-x) = - $\left\{\begin{array}{lll}1+x& when& -2<x<0\\ 1& when& 0\le x<2\end{array}\right.$

= $\left\{\begin{array}{lll}-1-x& when& -2<x<0\\ -1& when& 0\le x-2\end{array}\right.$ = F1(x)

Hence, option (c) is the correct option.

Alternate solution:
F1(2) = −1 = F(2) and F1(−2) = 1 = F(−2).

So the correct option is (c), i.e. F1(x) = − F(−x).

Hence, option (c).

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