Range restriction problems

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }\).

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote.
2. Show that \(1 < y \leqslant 3\) for all real values of \(x\).
3. Find the coordinates of any stationary points on \(C\).
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4. Sketch \(C\), stating the coordinates of any intersections with the axes and labelling the asymptote.
5. Sketch the curve with equation \(y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }\) and find the set of values of \(x\) for which \(\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }\).

7 A curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(0 < y < 8\). Sketch \(C\), giving the coordinates of the turning points.

10 A curve \(C\) has equation $$y = \frac { 5 \left( x ^ { 2 } - x - 2 \right) } { x ^ { 2 } + 5 x + 10 }$$ Find the coordinates of the points of intersection of \(C\) with the axes. Show that, for all real values of \(x , - 1 \leqslant y \leqslant 15\). Sketch \(C\), stating the coordinates of any turning points and the equation of the horizontal asymptote.
[0pt] [Question 11 is printed on the next page.]

8 A curve \(C\) has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$

1. State the equation of the asymptote of \(C\).
2. The line \(y = k\) intersects the curve \(C\). Show that \(4 k ^ { 2 } - 4 k - 3 \leqslant 0\).
3. Hence find the coordinates of the stationary points of the curve \(C\). (No credit will be given for solutions based on differentiation.)
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5

1. Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
2. Show that \(y\) cannot take values between - 3 and 1 .