Proving Rational Function Takes All Real Values

Show that a rational function can take all real values, typically by rearranging to a quadratic in x and showing the discriminant is always non-negative.

3 questions · Standard +0.8

1.02k Simplify rational expressions: factorising, cancelling, algebraic division 1.02n Sketch curves: simple equations including polynomials

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OCR FP2 2008 January Q6

8 marks Standard +0.8

6 The equation of a curve is \(y = \frac { 2 x ^ { 2 } - 11 x - 6 } { x - 1 }\).

Find the equations of the asymptotes of the curve.
Show that \(y\) takes all real values.

OCR FP2 2007 June Q9

11 marks Standard +0.8

9 It is given that the equation of a curve is $$y = \frac { x ^ { 2 } - 2 a x } { x - a }$$ where \(a\) is a positive constant.

Find the equations of the asymptotes of the curve.
Show that \(y\) takes all real values.
Sketch the curve \(y = \frac { x ^ { 2 } - 2 a x } { x - a }\).

OCR FP2 2011 June Q2

7 marks Standard +0.8

2 A curve has equation \(y = \frac { x ^ { 2 } - 6 x - 5 } { x - 2 }\).

Find the equations of the asymptotes.
Show that \(y\) can take all real values.