Discussion
The range of \(f(x) = [cosx]\) , where \([x]\) is the greatest integer less than or equal to \(x\) , is
Answer: D
The range is {-1, 0, 1}.
\(\cos x\) takes every value in \([-1, 1]\). The greatest integer function maps these values as follows.
\[ \begin{aligned} \cos x = 1 &\Rightarrow [\cos x] = 1 \\ 0 \le \cos x \lt 1 &\Rightarrow [\cos x] = 0 \\ -1 \le \cos x \lt 0 &\Rightarrow [\cos x] = -1 \end{aligned} \]
All three values occur (at x = 0, \(\frac{\pi}{2}\) and \(\pi\)), so the range is the set {-1, 0, 1}. The intervals (-1, 1) and [-1, 1] are wrong because \([\cos x]\) takes only integer values, and the two-element sets each miss one of the three.
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