First slide

Functions (XII)

Question

The function 'g' defined by $g (x) = \sin (\sin^{- 1} \{\sqrt{x}\}) + \cos (\sin^{- 1} \{\sqrt{x}\}) - 1$ where {x} denotes the functional part function is

Moderate

A

an even functioin
B

periodic function
C

odd function
D

Neither even nor odd

Solution

$\begin{array}{l} g \{x\} = \sin (\sin^{- 1} \sqrt{\{x\}}) + \cos (\sin^{- 1} \sqrt{\{x\}}) - 1 \\ = \sqrt{\{x\}} + \cos (\cos^{- 1} \sqrt{1 - \{x\}}) - 1 \\ = \sqrt{\{x\}} + \sqrt{1 - \{x\} - 1} \end{array}$
$\begin{array}{l} If x \in I then \{x\} = 0 then g (x) = 0 \\ If x \notin I then \{- x\} = 1 - \{x\} \\ \Rightarrow g (- x) = \sqrt{1 - \{x\}} + \sqrt{\{x\}} - 1 = g (x) \\ \Rightarrow g (- x) = g (x) \\ \Rightarrow g (x) is even function . \\ Also g (x) is periodic function \end{array}$

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