9.
$$\mathrm { f } ( x ) = 2 \ln ( x ) - 4 , \quad x > 0 , \quad x \in \mathbb { R }$$
- Sketch, on separate diagrams, the curve with equation
- \(y = \mathrm { f } ( x )\)
- \(y = | \mathrm { f } ( x ) |\)
On each diagram, show the coordinates of each point at which the curve meets or cuts the axes.
On each diagram state the equation of the asymptote.
- Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 4\)
$$\mathrm { g } ( x ) = \mathrm { e } ^ { x + 5 } - 2 , \quad x \in \mathbb { R }$$
- Find \(\mathrm { gf } ( x )\), giving your answer in its simplest form.
- Hence, or otherwise, state the range of gf.