(a)     Given that the domain of \(g\) is \(x \geqslant a\) , find the least value of \(a\) such that \(g\) has an inverse function.

(b)     On the same set of axes, sketch

  (i)     the graph of \(g\) for this value of \(a\) ;

  (ii)     the corresponding inverse, \({g^{ - 1}}\) .

(c)     Find an expression for \({g^{ - 1}}(x)\) .

(a)     \(a = 2.24\)     \(\sqrt 5 \)     A1

(b)     (i)

    A2

Note: Award A1 for end point

   A1 for its asymptote.

(ii)     sketch of \({g^{ - 1}}\) (see above)     A2

Note: Award A1 for end point

   A1 for its asymptote.

(c)     \(y = \frac{{3x}}{{5 + {x^2}}} \Rightarrow y{x^2} - 3x + 5y = 0\)     M1

\( \Rightarrow x = \frac{{3 \pm \sqrt {9 - 20{y^2}} }}{{2y}}\)     A1

\({g^{ - 1}}(x) = \frac{{3 \pm \sqrt {9 - 20{x^2}} }}{{2x}}\)     A1

[8 marks]

Very few completely correct answers were given to this question. Many students found a to be \(0\) and many failed to provide adequate sketches. There were very few correct answers to part (c) although many students were able to obtain partial marks.