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Answer the questions on the basis of the information given below:

  • $f_{1}(x) = \left\{\begin{matrix} x & 0 \leq x \leq 1 \\  1 & x \geq 1 \\ 0 & \text{otherwise} \end{matrix}\right.$  
  • $f_{2}(x) =  f_{1}(-x) \;\; \text{for all} \; x $
  • $f_{3}(x) =  -f_{2}(-x) \;\; \text{for all} \; x $
  • $f_{4}(x) =  f_{3}(-x) \;\; \text{for all} \; x $

 Which of the following is necessarily true?

  1. $f_4(x) = f_1(x) \: \text{for all }\;x$
  2. $f_1(x) = f_3(-x) \: \text{for all }\;x$
  3. $f_2(-x) = f_4(x) \: \text{for all }\;x$
  4. $f_1(x) + f_3(x) = 0 \: \text{for all }\;x$
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