Answer: sin x is periodic and not one-one over all reals, so it has no inverse without restriction.
- A sin x fails to be continuous at certain rational multiples of pi, breaking the inverse construction
- B sin x is periodic and not one-one over all reals, so it has no inverse without restriction
- C sin x has no defined range over the real numbers, so an inverse formula cannot be written
- D sin x is always positive for every real input value, leaving no negative outputs to invert
Correct answer: B. sin x is periodic and not one-one over all reals, so it has no inverse without restriction
Explanation: Since trig functions repeat periodically, they are not one-one on their full domain; restricting to a principal branch makes them invertible.
The graph of sin⁻¹x is confined to a narrow domain [-1,1] (since sine itself only takes values in that range) and range [-π/2,π/2] (the principal value branch chosen to make sine one-one and invertible there).
Concept context
Restricting trig functions to make them invertible, the principal value branches of sin-inverse, cos-inverse, tan-inverse, and friends, and the key identities relating them.