Find \({h^{ - 1}}(x)\) .

(i)     Sketch the graph of h for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any asymptotes.

(ii)    Write down the equations of the asymptotes.

(iii)   Write down the x-intercept of the graph of h .

Let R be the region in the first quadrant enclosed by the graph of h , the x-axis and the line \(x = 3\).

(i)     Find the area of R.

(ii)    Write down an expression for the volume obtained when R is revolved through \({360^ \circ }\) about the x-axis.

\(y = \frac{{2x - 1}}{{x + 1}}\)

interchanging x and y (seen anywhere)     M1

e.g. \(x = \frac{{2y - 1}}{{y + 1}}\)

correct working     A1

e.g. \(xy + x = 2y - 1\)

collecting terms     A1

e.g. \(x + 1 = 2y - xy\) , \(x + 1 = y(2 - x)\)

\({h^{ - 1}}(x) = \frac{{x + 1}}{{2 - x}}\)     A1     N2

[4 marks]

     A1A1A1A1     N4

Note: Award A1 for approximately correct intercepts, A1 for correct shape, A1 for asymptotes, A1 for approximately correct domain and range.

(ii) \(x = - 1\) , \(y = 2\)     A1A1     N2

(iii) \(\frac{1}{2}\)     A1     N1

[7 marks]

(i) \({\text{area}} = 2.06\)     A2     N2

(ii) attempt to substitute into volume formula (do not accept \(\pi \int_a^b {{y^2}{\rm{d}}x} \) )     M1

volume \( = \pi {\int_{\frac{1}{2}}^3 {\left( {\frac{{2x - 1}}{{x + 1}}} \right)} ^2}{\rm{d}}x\)     A2     N3

[5 marks]