Problem

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3+6x-2x^2$ for $x\in\mathbb{R}$.

(a) Express $\mathrm{f}(x)$ in the form $a-b(x-c)^2$, where $a$, $b$ and $c$ are constants, and state the range of $\mathrm{f}$.

(b) The graph of $y=\mathrm{f}(x)$ is transformed to the graph of $y=\mathrm{h}(x)$ by a reflection in one of the axes followed by a translation. It is given that the graph of $y=\mathrm{h}(x)$ has a minimum point at the origin.

Give details of the reflection and translation involved.

The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=3+6x-2x^2$ for $x\leq 0$.

(c) Sketch the graph of $y=\mathrm{g}(x)$ and explain why $\mathrm{g}$ is a one-one function. You are not required to find the coordinates of any intersections with the axes.

(d) Sketch the graph of $y=\mathrm{g}^{-1}(x)$ on your diagram in (c), and find an expression for $\mathrm{g}^{-1}(x)$. You should label the two graphs in your diagram appropriately and show any relevant mirror line.