Sketch the graph of \(y = f(x)\), indicating clearly any asymptotes and points of intersection with the \(x\) and \(y\) axes.

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

Find all values of \(x\) for which \(f(x) = {f^{ - 1}}(x)\).

Solve the inequality \(\left| {f(x)} \right| < \frac{3}{2}\).

Solve the inequality \(f\left( {\left| x \right|} \right) < \frac{3}{2}\).

Note: In the diagram, points marked \(A\) and \(B\) refer to part (d) and do not need to be seen in part (a).

shape of curve     A1

Note:     This mark can only be awarded if there appear to be both horizontal and vertical asymptotes.

intersection at \((0,{\text{ }}0)\)     A1

horizontal asymptote at \(y = 3\)     A1

vertical asymptote at \(x = 2\)     A1

[4 marks]

\(y = \frac{{3x}}{{x - 2}}\)

\(xy - 2y = 3x\)     M1A1

\(xy - 3x = 2y\)

\(x = \frac{{2y}}{{y - 3}}\)

\(\left( {{f^{ - 1}}(x)} \right) = \frac{{2x}}{{x - 3}}\)     M1A1

Note:     Final M1 is for interchanging of \(x\) and \(y\), which may be seen at any stage.

[4 marks]

METHOD 1

attempt to solve \(\frac{{2x}}{{x - 3}} = \frac{{3x}}{{x - 2}}\)     (M1)

\(2x(x - 2) = 3x(x - 3)\)

\(x\left[ {2(x - 2) - 3(x - 3)} \right] = 0\)

\(x(5 - x) = 0\)

\(x = 0\;\;\;\)or\(\;\;\;x = 5\)     A1A1

METHOD 2

\(x = \frac{{3x}}{{x - 2}}\;\;\;\)or\(\;\;\;x = \frac{{2x}}{{x - 3}}\)     (M1)

\(x = 0\;\;\;\)or\(\;\;\;x = 5\)     A1A1

[3 marks]

METHOD 1

at \({\text{A}}:\frac{{3x}}{{x - 2}} = \frac{3}{2}\) AND at \({\text{B}}:\frac{{3x}}{{x - 2}} =  - \frac{3}{2}\)     M1

\(6x = 3x - 6\)

\(x =  - 2\)     A1

\(6x = 6 - 3x\)

\(x = \frac{2}{3}\)     A1

solution is \( - 2 < x < \frac{2}{3}\)     A1

METHOD 2

\({\left( {\frac{{3x}}{{x - 2}}} \right)^2} < {\left( {\frac{3}{2}} \right)^2}\)     M1

\(9{x^2} < \frac{9}{4}{(x - 2)^2}\)

\(3{x^2} + 4x - 4 < 0\)

\((3x - 2)(x + 2) < 0\)

\(x =  - 2\)     (A1)

\(x = \frac{2}{3}\)     (A1)

solution is \( - 2 < x < \frac{2}{3}\)     A1

[4 marks]

\( - 2 < x < 2\)     A1A1

Note:     A1 for correct end points, A1 for correct inequalities.

Note:     If working is shown, then A marks may only be awarded following correct working.

[2 marks]

Total [17 marks]