N2010 P2 Q4

The function \(\mathrm f\) is defined as follows: \(\mathrm f:x\mapsto\frac1{x^2-1}\) for \(x\in\mathbb R\), \(x\ne-1\), \(x\ne1\).

1. Sketch the graph of \(y=\mathrm f(x)\).[1]
2. If the domain of \(\mathrm f\) is further restricted to \(x\ge k\), state with a reason the least value of \(k\) for which the function \(\mathrm f^{-1}\) exists.[2]

In the rest of the question, the domain of \(\mathrm f\) is \(x\in\mathbb R\), \(x\ne-1\), \(x\ne1\), as originally defined.

The function \(\mathrm g\) is defined as follows: \(\mathrm g:x\mapsto\frac1{x-3}\) for \(x\in\mathbb R\), \(x\ne2\), \(x\ne3\), \(x\ne4\).

3. Show that \(\mathrm{fg}(x)=\frac{(x-3)^2}{(4-x)(x-2)}\).[2]
4. Solve the inequality \(\mathrm{fg}(x)>0\).[3]
5. Find the range of \(\mathrm{fg}\).[3]