Which of the following functions have inverse defined on their ranges ?

The function in (a) is not one-one, so f−1  is not defined. The function in (b) is one-one and onto defined from R→R,  hence f−1  is defined from R→R given by f−1(x)=x1/3. The function f(x)=sinx  is not one-one in (0,2π) as f(π/4)=f(3π/4)=1/2,  so it is not invertible.The function f(x)=ex  is one-one as ex=ey⇔x=y. The range f=R+={x:x>0}. So inverse of f ie logx  is defined on R+  to R.