2020 DHS Promo Q3

Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by

\(\mathrm{f}:x\mapsto {{x}^{2}}+cx+d,x\in \mathbb{R},\) where \(c\) and \(d\) are constants,

\(\mathrm{g}:x\mapsto \ln x,x\in \mathbb{R},x>0\).

1. Given that the composite function \(\mathrm{gf}\) exists, find an inequality involving \(c\) and \(d\).[2]

For the rest of the question, take \(c=2,d=4\).

2. Write down an expression for \(\mathrm{gf}\left( x \right)\) and find exactly the range of \(\mathrm{gf}\).[3]
3. The graph of a function \(\mathrm{h}\) is symmetrical about the line \(x=k\) if \(\mathrm{h}\left( x+k \right)=\mathrm{h}\left( -x+k \right)\) for all valid values of \(x\). Using this definition, show that the graph of \(\mathrm{gf}\) is symmetrical about the line \(x=-1\).[2]