First slide

Functions (XII)

Question

$Let f_{1} (x) = \{\begin{cases} x, 0 \leq x \leq 1 \\ 1, x > 1 \\ 0, otherwise \end{cases}$
$f_{2} (x) = f_{1} (- x) for all x$
$f_{3} (x) = - f_{2} (x) for all x$
$f_{4} (x) = f_{3} (- x) for all x$
which of the following is necessarily true?

Moderate

A

$f_{4} (x) = f_{1} (x) for all x$
B

$f_{1} (x) = - f_{3} (- x) for all x$
C

$f_{2} (- x) = f_{4} (x) for all x$
D

$f_{1} (x) + f_{3} (x) = 0 for all x$

Solution

$f_{2} (x) = f_{1} (- x) f_{3} (x) = - f_{2} (x) f_{3} (x) = - f_{1} (- x) r e p l a c e x b y - x f_{3} (- x) = - f_{1} (x) \Rightarrow f_{1} (x) = - f_{3} (- x)$

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