Reconstruct a transformation from its intervals

8 written marks

A function \(f\) has domain \([-2,4]\) and range \([1,7]\). The graph includes \((1,7)\), and \(f(u)=1\) exactly when \(u=-2\) or \(u=4\). Let \(g(x)=A f(Bx+C)+D\), where \(A<0\), \(B>0\). The domain of \(g\) is \([1,3]\) and its range is \([-9,3]\).

Complete-answer checking, worked solutions and indicative written criteria.