Functions

Step 1: Analyze the denominator The given integral has the denominator: Using trigonometric identities,

Step 2: Domain restrictions The function is undefined where the denominator is zero: Solving for , Thus, the domain of is:

Step 1: Checking the invertibility of
The function is a quadratic function. A function is invertible if it is one-to-one (injective). However, since quadratic functions are not one-to-one over , is not invertible.

Step 2: Checking if is an identity function
The identity function is defined as . However, the given function is:

This does not satisfy for all , so it is not an identity function.

Step 3: Checking if is invertible
Since is not a one-to-one function over , is not invertible.

Step 4: Checking if
From the definition of , we get:

Since for all , we always have . Hence, holds true. Since statements (III) and (IV) are correct, the correct answer is (C).