True or false: existence of inverses

Exercise and reference answers

Questions

1. Statement is true

   Yes

   Statement

   A function has an inverse only when it is one-one and onto.

2. Statement is true

   No

   Statement

   sin x is one-one over all of R, so sin⁻¹ needs no restriction.

3. Statement is true

   Yes

   Statement

   Restricting sine to [−π/2, π/2] makes it one-one with range [−1, 1].

4. Statement is true

   No

   Statement

   sin⁻¹ x means the same thing as (sin x)⁻¹.

5. Statement is true

   Yes

   Statement

   (sin x)⁻¹ is 1/sin x, which is cosec x.

6. Statement is true

   No

   Statement

   cos⁻¹ is defined for every real number.

Explanation