(i) Express $x^2-2x-15$ in the form $(x+a{)}^2+b$. The function $f$ is defined for $p⩽x⩽q$, where $p \& q$ are positive constants, by $\mathrm{f}:x\mapsto x^2-2x-15$.

The range of $\mathrm{f}$ is given by $c⩽\mathrm{f}\left(x\right)⩽d$, where $c \& d$ are constants.

(ii) State the smallest possible value of $c$.

For the case where $c=9$ and $d=65$,

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 2x^2-12x+7$ for $x\in \mathrm{ℝ}$.

(i) Express $f\left(x\right)$ in the form $a(x-b{)}^2-c$.

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 2x^2-12x+7$ for $x\in \mathrm{ℝ}$.

(ii) State the range of $\mathrm{f}$.

The function $\mathrm{f}$(x) is defined by $\mathrm{f}:x\mapsto 2x^2-12x+7$ for $x\in \mathrm{ℝ}$.

Find the set of values of $x$ for which $f\left(x\right)<21$.

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 2x^2-12x+7$ for $x\in \mathrm{ℝ}$.

(iv) Find the value of the constant $k$ for which the equation $\mathrm{gf}\left(x\right)=0$ has two equal roots.

12 Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x\in \mathrm{ℝ}$ by

$\mathrm{f}:x\mapsto 2x+1$

$\mathrm{g}:x\mapsto x^2-2$

(i) Find and simplify expressions for $fg\left(x\right)$ and $\mathrm{gf}\left(\mathrm{x}\right)$.