Consider the function \(f(x)=\frac{x^2+bx+c}{x+d}\), where \(x\in\mathbb{R}\), \(x\ne-d\), and \(b,c,d\in\mathbb{Z}\).
The graph of \(f\) has asymptotes \(x=3\) and \(y=x+7\).
Find the value of \(d\).[1]
Show that \(b=4\).[2]
The graph of \(f\) has a local maximum when \(x=-3\).
Find the value of \(c\).[4]
Sketch the graph of \(f\) for \(-30\leq x\leq30\), clearly indicating any intersections with the axes, local maximum and minimum points, and asymptotes.[4]
The equation \(\lvert f(x)\rvert=k\) has four distinct real solutions.
Find the range of possible values of \(k\).[2]
The graph of \(g\) is obtained by translating the graph of \(f\) by the vector \(\begin{pmatrix}2\\3\end{pmatrix}\).
Find the equations of the vertical asymptotes of the graph of \(y=\frac1{g(x)}\).[4]