The number of integers in the domain of the function \(f(x) = \sin^{-1}\left(\displaystyle\frac{x - 3}{2}\right)\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
\(\sin^{-1} t\) is defined only for \(-1 \le t \le 1\). So the argument must lie in \([-1, 1]\):
\( \displaystyle -1 \le \frac{x - 3}{2} \le 1 \)
Multiplying by 2:
\(-2 \le x - 3 \le 2\)
Adding 3:
\(1 \le x \le 5\)
The domain is \([1, 5]\). Both end points are included, so the integers in it are 1, 2, 3, 4 and 5, that is, 5 integers.
Why the other options are incorrect
Option (1) 3 — Incorrect. This counts only 2, 3, 4 and leaves out both end points 1 and 5, which are in the domain.
Option (2) 4 — Incorrect. This is the length \(5 - 1\) of the interval, not the number of integers in it.
Option (4) 6 — Incorrect. There is no sixth integer: \(x = 0\) gives \(\frac{-3}{2}\) and \(x = 6\) gives \(\frac{3}{2}\), both outside \([-1, 1]\).