1.08f Area between two curves: using integration

Two curves are defined by \(y = x^k\) and \(y = x^{\frac{1}{k}}\), for \(x \geqslant 0\), where \(k > 0\).

1. Prove that, except for one value of \(k\), the curves intersect in exactly two points. [4]

The two curves enclose a finite region \(R\).

1. Find the area, \(A\), of \(R\), giving your answer in the form \(A = f(k)\) and distinguishing clearly between the cases \(k < 1\) and \(k > 1\). [4]
2. Determine the set of values of \(k\) for which \(A \leqslant 0.5\). [3]
3. The function \(f\) is given by \(f : x \mapsto A\) with \(k > 1\). Prove that \(f\) is one-one and determine its inverse. [4]