Inverse transformation or reflection

7 The diagram shows a part \(A B C\) of the curve \(y = 3 - 2 x ^ { 2 }\), together with its reflections in the lines \(y = x\), \(y = - x\) and \(y = 0\). \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-05_691_673_1957_678}

4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\) and f is a one-one function.

1. Describe a single transformation that transforms \(C\) to the curve with equation \(y = - \mathrm { f } ( - x )\) . The curve \(C\) is reflected in the line with equation \(y = - x\) to give the curve \(V\) . The equation of \(V\) is \(y = \mathrm { g } ( x )\) .
2. Explain why \(\mathrm { g } ^ { - 1 } ( x ) = - \mathrm { f } ( - x )\) . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-3_793_979_819_633} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 3 ( x - 1 ) } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ The curve has asymptotes with equations \(x = p\) and \(y = q\) and \(C\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\) .
3. Write down the value of \(p\) and the value of \(q\) .
4. Write down the coordinates of the point \(A\) and the coordinates of the point \(B\) . Given the definition of \(\mathrm { g } ( x )\) in part(b),
5. find the function g .
6. Solve \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = x\)

The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\). The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\) [2 marks]