Inverse and composite of linear functions, a line from its intercepts and the reflection in y equals x

The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=3x-4\), \(x\in\mathbb{R}\).

1. Find
   1. \(\mathrm{f}(5)\)
   2. \(\mathrm{f}^{-1}(x)\).[3]

The diagram shows part of the straight-line graph of a function \(\mathrm{h}\). The line crosses the \(x\)-axis at \(x=-3\) and the \(y\)-axis at \(y=2\).

2. Write down \(\mathrm{h}(x)\) in terms of \(x\).[2]
3. Solve \(\mathrm{h}^{-1}(x)=3\).[2]

The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=mx+c\), \(x\in\mathbb{R}\), where \(m\) and \(c\) are rational constants.

4. Given that \(\mathrm{h}(x)=(\mathrm{f}^{-1}\circ\mathrm{g})(x)\) for all \(x\), find \(m\) and \(c\).[4]

The function \(\mathrm{k}\) satisfies \(\mathrm{h}(\mathrm{k}(x))=x\) for all \(x\).

5. Find \(\mathrm{k}(x)\) in terms of \(x\).[2]
6. State the single geometric transformation which takes the graph of \(y=\mathrm{k}(x)\) to the graph of \(y=\mathrm{h}(x)\).[1]