Proving Excluded Range of Rational Function

Show that a rational function cannot take values in a specific interval (e.g., prove there are no points for a < y < b) using discriminant or algebraic argument.

3 questions · Standard +0.9

1.02n Sketch curves: simple equations including polynomials
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OCR FP2 2014 June Q5
9 marks Standard +0.8
5 A curve has equation \(y = \frac { x ^ { 2 } - 8 } { x - 3 }\).
  1. Find the equations of the asymptotes of the curve.
  2. Prove that there are no points on the curve for which \(4 < y < 8\).
  3. Sketch the curve. Indicate the asymptotes in your sketch.
CAIE FP1 2014 June Q12 OR
Challenging +1.2
The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$ where \(a , b , c\) and \(d\) are constants. The curve cuts the \(y\)-axis at \(( 0 , - 2 )\) and has asymptotes \(x = 2\) and \(y = x + 1\).
  1. Write down the value of \(d\).
  2. Determine the values of \(a , b\) and \(c\).
  3. Show that, at all points on \(C\), either \(y \leqslant 3 - 2 \sqrt { 6 }\) or \(y \geqslant 3 + 2 \sqrt { 6 }\).
CAIE FP1 2014 November Q4
7 marks Standard +0.8
4 A curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }\). Find the equations of the asymptotes of \(C\). Show that there is no point on \(C\) for which \(1 < y < 9\).