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]