6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }\).
- Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote.
- Show that \(1 < y \leqslant 3\) for all real values of \(x\).
- Find the coordinates of any stationary points on \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-12_2718_42_107_2007}
\includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-13_2720_40_106_18} - Sketch \(C\), stating the coordinates of any intersections with the axes and labelling the asymptote.
- Sketch the curve with equation \(y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }\) and find the set of values of \(x\) for which \(\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }\).