The domain of the function \(\cos^{-1}(2x-3)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → [1, 2]
\(\cos^{-1} u\) is defined only for \(-1 \le u \le 1\), the closed interval [−1, 1]. So
\(-1 \le 2x - 3 \le 1\)
\(2 \le 2x \le 4\)
\(1 \le x \le 2\)
Why the other options are incorrect
Option (2) [-1, 1] — Incorrect. This is the interval for \(2x - 3\), not for x. For example, x = 0 gives \(\cos^{-1}(-3)\), which is not defined.
Option (3) (1, 2) — Incorrect. It leaves out x = 1 and x = 2, where \(2x - 3 = -1\) and 1, and \(\cos^{-1}(-1) = \pi\), \(\cos^{-1}(1) = 0\) are defined.
Option (4) (-1, 1) — Incorrect. For every x in it, \(2x - 3 \lt -1\), so \(\cos^{-1}(2x - 3)\) is not defined there at all.