First slide

Functions (XII)

Question

$T h e r e a l v a l u e d f u n c t i o n$ $f (x) = \frac{\cos e c^{- 1} x}{\sqrt{x - [x]}}$ , $w h e r e [x] d e n o t e s t h e g r e a t e s t i n t e g e r l e s s t h a n o r e q u a l t o x, i s d e f i n e d f o r a l l x b e l o n g s t o$

Moderate

A

R $-$ Z
B

$all integers except \{0, - 1, 1\}$
C

$R-Z ∪ [- 1, 1]$
D

$all reals except the interval [- 1, 1]$

Solution

$\begin{array}{l} f (x) = \frac{\cos e c^{- 1} (x)}{\sqrt{x - [x]}} \\ \cos e c^{- 1} x i s d e f i n e d f o r a l l | x | \geq 1 \\ x \leq - 1 or x \geq 1 \\ x \neq [x] \Rightarrow x d o e s n o t i n t e g r a l v a l u e s \\ ∴ x \notin z and x \notin [- 1, 1] \end{array}$

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