First slide

Functions (XII)

Question

$The number of integral elements in the domain of the function$
$f (x) = \sin^{- 1} (\frac{x^{2} - 2 x}{3}) + \sqrt{[x] + [- x]}, where [.]denotes greatest integer function, is$

Moderate

A

6
B

4
C

infinite
D

5

Solution

$\begin{array}{l} Given that f (x) = \sin^{- 1} (\frac{x^{2} - 2 x}{3}) + \sqrt{[x] + [- x]} \\ Now, \sqrt{[x] + [- x]} is defined only for integral values of ' x' \\ And \sin^{- 1} (\frac{x^{2} - 2 x}{3}) is defined when - 1 \leq \frac{x^{2} - 2 x}{3} \leq 1 \\ If \frac{x^{2} - 2 x}{3} \geq - 1 \\ \Rightarrow x^{2} - 2 x + 3 \geq 0 \\ D < 0 \end{array}$
$It has no real values$
$and i f \frac{x^{2} - 2 x}{3} \leq 1$
$\begin{array}{l} \Rightarrow x^{2} - 2 x - 3 \leq 0 \\ \Rightarrow (x + 1) (x - 3) \leq 0 \\ \Rightarrow x \in [- 1, 3] \\ ∴ Domain of f (x) is [- 1, 3] \\ ∴ x \in {- 1, 0, 1, 2, 3} (∵ Integralvalues only) \end{array}$
$∴ Number of integral elements in the domain are 5$

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