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Asymptotes and axis intercepts of a reciprocal-type rational function, then its sketch

Asymptotes and axis intercepts of a reciprocal-type rational function, then its sketch

5 marks

The function \(\mathrm{f}\) is given by \(\mathrm{f}(x)=2-\frac{4}{x+1}\), for \(x\in\mathbb{R}\), \(x\neq-1\).

  1. State an equation for the curve's
    1. vertical asymptote;
    2. horizontal asymptote.[2]
  2. Find where the curve \(y=\mathrm{f}(x)\) crosses each axis, giving the coordinates of the point on
    1. the \(y\)-axis;
    2. the \(x\)-axis.[2]
  3. Using the axes below, sketch the curve \(y=\mathrm{f}(x)\). Show the asymptotes and the axis intercepts found in parts (a) and (b).[1]

Solution:

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Answer:(a)(i) \(x=-1\) (a)(ii) \(y=2\) (b)(i) \((0,-2)\) (b)(ii) \((1,0)\) (c) two branches with asymptotes \(x=-1\) and \(y=2\), the right-hand branch through \((0,-2)\) and \((1,0)\)

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