(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.