FunctionMedium
Question
Let [ •] denote the greatest integer function. If the domain of the function $f(x) = \cos^{- 1}\left( \frac{4x + 2\lbrack x\rbrack}{3} \right)$ is $\lbrack\alpha,\beta\rbrack$, then $12(\alpha + \beta)$ is equal to :
Options
A.6
B.8
C.9
D.4
Solution
Sol. [Key Concept: The $\cos^{- 1}$ argument must lie in $\lbrack - 1,1\rbrack$; solve the greatest-integer inequality to get $\lbrack\alpha,\beta\rbrack$.]
$${- 1 \leq \frac{4 \times + 2\lbrack x\rbrack}{3} \leq 1 }{- 3 - 4x \leq 2\lbrack x\rbrack \leq 3 - 4x }$$
$$\begin{matrix} & x \in \left\lbrack \frac{- 1}{4},\frac{3}{4} \right\rbrack \\ & 12(\alpha + \beta) = 6 \end{matrix}$$
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