The number of integers in the range of the function \(f(x) = 3\cos^{-1} x\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 10
As x runs over \([-1, 1]\), \(\cos^{-1} x\) takes every value in \([0, \pi]\), including both ends. Multiplying by 3:
\(0 \le 3\cos^{-1} x \le 3\pi\)
So the range is \([0, 3\pi]\). Taking \(\pi \approx 3.14\),
\(3\pi \approx 9.42\)
The integers in \([0, 9.42]\) are \(0, 1, 2, \ldots, 9\), that is 10 integers.
Why the other options are incorrect
Option (1) 9 — Incorrect. It leaves out 0, which is in the range: \(f(1) = 3\cos^{-1} 1 = 0\).
Option (2) 4 — Incorrect. This counts the integers 0, 1, 2, 3 in \([0, \pi]\), the range of \(\cos^{-1} x\) before multiplying by 3.
Option (3) 11 — Incorrect. It includes 10, but \(10 \gt 3\pi \approx 9.42\).