2023 SJI P1 Q8

Consider the function

\[\mathrm f(x)=\begin{cases}x,&x<0,\\ x+1,&x\geq0.\end{cases}\]

1. Find \(\mathrm f(3)\).[1]
2. 1. Show that \(\mathrm f\) is one-to-one.
   2. State the range of \(\mathrm f\).
   3. Find the inverse function, \(\mathrm f^{-1}\) and its domain, giving your answer in similar form.[5]
3. Sketch the graph of \(y=(\mathrm f\circ\mathrm f^{-1})(x)\).[3]

It is known that for a composite function \(\mathrm h\circ\mathrm k\) to exist, the range of the function \(\mathrm k\) must be a subset of the domain of the function \(\mathrm h\).

Consider another function \(\mathrm g(x)=\sqrt x\), \(x\geq c\) where \(c\) is a constant.

4. Find the least value of \(c\) for which the composite function \(\mathrm f^{-1}\circ\mathrm g\) exists.[2]
5. Given that \(\mathrm f^{-1}\circ\mathrm g\) exists and \((\mathrm f^{-1}\circ\mathrm g)(x)=3\), find the value of \(x\).[2]