The functions $f$ and $g$ are given by $f(x) = 2x^2 + 8x + 1$ for $x \in \mathbb{R}$, and $g(x) = 2x - k$ for $x \in \mathbb{R}$, where $k$ is a constant.
(i)[3]
Determine the value of $k$ so that the line $y = g(x)$ touches the curve $y = f(x)$.
(ii)[3]
When $k = -9$, determine the set of $x$ values for which $f(x) < g(x)$.
(iii)[3]
When $k = -1$, find $g^{-1}(x)$ and solve $g^{-1}f(x) = 0$.
(iv)[3]
Write $f(x)$ in the form $2(x + a)^2 + b$, where $a$ and $b$ are constants, and hence state the least value of $f(x)$.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Forms a quadratic equation and applies the discriminant $b^2-4ac=0$” …