Question
Let [ • ] denote the greatest integer function. If the domain of the function $f(x) = \sin^{- 1}\left( \frac{x + \lbrack x\rbrack}{3} \right)$ is $\lbrack\alpha,\beta)$, then $\alpha^{2} + \beta^{2}$ is equal to:
Options
Solution
Sol. [Key Concept: The domain of $\sin^{- 1}$ needs $- 1 \leq \frac{x + \lbrack x\rbrack}{3} \leq 1$; solve this greatest-integer inequality piecewise over each integer value of $\lbrack x\rbrack$ to get the interval $\lbrack\alpha,\beta)$.]
$${- 1 \leq \frac{x + \lbrack x\rbrack}{3} \leq 1 }{\Rightarrow - 3 \leq x - 1\lbrack x\rbrack \leq 3 }{\Rightarrow - 3 - \lbrack x\rbrack \leq x \leq 3 - \lbrack x\rbrack }$$if $\lbrack x\rbrack = - 1 \Rightarrow x \in \lbrack - 1,0)$
if $\lbrack x\rbrack = 0 \Rightarrow x \in \lbrack 0,1)$
if $\lbrack x\rbrack = 1 \Rightarrow x \in \lbrack 1,2)$
Hence $x \in \lbrack - 1,2)$
$\Rightarrow \alpha = - 1$ and $\beta = 2$
$$\Rightarrow \alpha^{2} + \beta^{2} = 5 $$
Create a free account to view solution
View Solution Free