For $x \in \mathbb{R}$, the functions $f$ and $g$ are given by $f : x \mapsto \frac{1}{2}x - 2$, $g : x \mapsto 4 + x - \frac{1}{2}x^2$.
(i)[3]
Find where the graphs of $y = f(x)$ and $y = g(x)$ intersect.
(ii)[2]
Find the set of $x$ values for which $f(x) > g(x)$.
(iii)[4]
Obtain an expression for $fg(x)$ and deduce the range of $fg$.
(iv)[2]
The function $h$ is given by $h : x \mapsto 4 + x - \frac{1}{2}x^2$ for $x \ge k$. Find the smallest value of $k$ such that $h$ has an inverse.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Sets $f(x)$ equal to $g(x)$ and obtains $x^2-x-12=0$.” …