Find the inverse function ${f^{ - 1}}$, stating its domain.

Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.

Sketch the graph of $y = g\left( x \right)$. State the equations of any asymptotes and the coordinates of any intercepts with the axes.

The function $h$ is defined by $h\left( x \right) = \sqrt x$, for $x$ ≥ 0.

State the domain and range of $h \circ g$.

attempt to make $x$ the subject of $y = \frac{{ax + b}}{{cx + d}}$      M1

$y\left( {cx + d} \right) = ax + b$      A1

$x = \frac{{dy - b}}{{a - cy}}$     A1

${f^{ - 1}}\left( x \right) = \frac{{dx - b}}{{a - cx}}$     A1

Note: Do not allow $y =$ in place of ${f^{ - 1}}\left( x \right)$.

$x \ne \frac{a}{c},\,\,\,\left( {x \in \mathbb{R}} \right)$     A1

Note: The final A mark is independent.

[5 marks]

$g\left( x \right) = 2 + \frac{1}{{x - 2}}$     A1A1

[2 marks]

hyperbola shape, with single curves in second and fourth quadrants and third quadrant blank, including vertical asymptote $x = 2$     A1

horizontal asymptote $y = 2$     A1

intercepts $\left( {\frac{3}{2},\,0} \right),\,\left( {0,\,\frac{3}{2}} \right)$     A1

[3 marks]

the domain of $h \circ g$ is $x \leqslant \frac{3}{2},\,\,x > 2$     A1A1

the range of $h \circ g$ is $y \geqslant 0,\,\,y \ne \sqrt 2$     A1A1

[4 marks]