2019 HCI Promo Q10

Describe a sequence of transformations which transforms the graph of \(y=\frac{1}{x}\) to the graph of \(y=\frac{3x+5}{x+2}\).[2]

Sketch the graphs of \(y=\frac{3x+5}{x+2}\)and \(16{{\left( x+3 \right)}^{2}}+9{{\left( y-4 \right)}^{2}}=144\) on a single diagram, indicating clearly any axial intercepts, points of intersection of the two graphs and the equations of asymptotes.[5]

Hence find the set of values of \(x\) that satisfies the inequality

\(\frac{3x+5}{x+2}>4-\sqrt{\frac{144-16{{\left( x+3 \right)}^{2}}}{9}}\) .[2]