11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-16_572_1338_278_239}
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\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\)
The curve meets the coordinate axes at the points \(A ( 0 , - 3 )\) and \(B \left( - \frac { 1 } { 3 } \ln 4,0 \right)\) and the curve has an asymptote with equation \(y = - 4\)
In separate diagrams, sketch the graph with equation
- \(y = | f ( x ) |\)
- \(y = 2 \mathrm { f } ( x ) + 6\)
On each sketch, give the exact coordinates of the points where the curve crosses or meets the coordinate axes and the equation of any asymptote.
Given that
$$\begin{array} { l l }
\mathrm { f } ( x ) = \mathrm { e } ^ { - 3 x } - 4 , & x \in \mathbb { R }
\mathrm {~g} ( x ) = \ln \left( \frac { 1 } { x + 2 } \right) , & x > - 2
\end{array}$$ - state the range of f,
- find \(\mathrm { f } ^ { - 1 } ( x )\),
- express \(f g ( x )\) as a polynomial in \(x\).