Suppose you wanted to give cosine function an inverse by restricting it's domain to [pi, k] for some value of k. what should the value of k be?

inverse of cosine function is cos^-1 x

if we restrict the domain of cosine function

then , value of k would be 2pi

For the inverse of a function to exist,

The function must be one to one & onto.

For the values more than π

Means the cosine goes into third and fourth quadrant.

Now in the third quadrant, cosine is negative.

Also in the fourth quadrant cosine is positive.

So for any value of cosine of a value for the third and fourth quadrant, we shall have a unique value.

Hence the cosine function shall be one to one in the third & fourth quadrant.

Also all values from -1 to 1 shall be taken by cosine.

So it shall also be onto.

Hence the inverse of cosine shall exist from π to 2π.

Hence the value of k shall be 2π