- The functions \(f\) and \(g\) are defined by
\[\begin{align*}
f&: x\mapsto e^x, &&x\in\mathbb{R}, \\
g&: x\mapsto x-1, &&x\in\mathbb{R}.
\end{align*}\]
Sketch in a single diagram the graphs of \(f\) and the composite functions \(gf\) and \(fg\), labelling each graph clearly.

State briefly the relationship

between the graphs of \(f\) and \(gf\),

between the graphs of \(f\) and \(fg\).

Express \((1-x)(x-3)\) in the form \(a-(x-b)^2\), where \(a\) and \(b\) are constants. State the coordinates of the maximum point on the graph of \(y=(1-x)(x-3)\), and state what symmetry the graph possesses.

Sketch the graph of \(y=e^{(1-x)(x-3)}\).