Question
Let
Let
\[\begin{array}{l}
{f_1}\left( x \right) = \left\{ \begin{array}{l}
x,\,\,\,0 \le x \le 1\\
1,\,\,\,x > 1\\
0,\,\,\,{\rm{otherwise}}
\end{array} \right.\\
{f_2}\left( x \right) = {f_1}\left( { - x} \right){\rm{ for\,\,all\,\, }}x\\
{f_3}\left( x \right) = - {f_2}\left( x \right){\rm{ for\,\,all\,\, }}x\\
{f_4}\left( x \right) = {f_3}\left( { - x} \right){\rm{ for\,\, all\,\, }}x
\end{array}\]
Which of the following is necessarily true ?
A.
$${f_4}\left( x \right) = {f_1}\left( x \right){\text{ for all }}x$$
B.
$${f_1}\left( x \right) = - {f_3}\left( { - x} \right){\text{ for all }}x$$
C.
$${f_2}\left( { - x} \right) = {f_4}\left( x \right){\text{ for all }}x$$
D.
$${f_1}\left( x \right) + {f_3}\left( x \right) = 0{\text{ for all }}x$$
Answer :
$${f_1}\left( x \right) = - {f_3}\left( { - x} \right){\text{ for all }}x$$
Solution :
