N2012 P1 Q7

A function \(f\) is said to be self-inverse if \(f(x)=f^{-1}(x)\) for all \(x\) in the domain of \(f\).

The function \(g\) is defined by \(g:x\mapsto\frac{x+k}{x-1}\), \(x\in\mathbb R\), \(x\ne1\), where \(k\) is a constant, \(k\ne-1\).

1. Show that \(g\) is self-inverse.[2]
2. Given that \(k>0\), sketch the curve \(y=g(x)\), stating the equations of any asymptotes and the coordinates of any points where the curve crosses the \(x\)- and \(y\)-axes.[3]
3. State the equation of one line of symmetry of the curve in part (ii), and describe fully a sequence of transformations which would transform the curve \(y=1/x\) onto this curve.[4]