The functions $f$ and $g$ are specified by $f : x \mapsto 2x + 5$ for $x \in \mathbb{R}$, and $g : x \mapsto \frac{8}{x - 3}$ for $x \in \mathbb{R}, x \neq 3$.
(i)[4]
Obtain $f^{-1}(x)$ and $g^{-1}(x)$ as formulas in $x$, and state the value of $x$ for which $g^{-1}(x)$ is not defined.
(ii)[3]
Sketch the graphs of $y = f(x)$ and $y = f^{-1}(x)$ on one diagram, and make the relationship between the two graphs clear.
(iii)[5]
For the equation $fg(x) = 5 - kx$, with $k$ a constant, having no solutions, find the possible values of $k$.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Correctly gives $f^{-1}(x)=\frac{1}{2}(x-5)$” …