For every real value of $x$, the functions f and g are defined by $f(x) = 4x^2 - c$ and $g(x) = 2x + k$, where $c$ and $k$ are positive constants. It is stated that $g^{-1}(3k + 1) = c$.
(a)[4]
Show that $gf(x) = 8x^2 - k - 1$ is obtained.
(b)[3]
Starting from the curve with equation $y = 8x^2 - k - 1$, it is transformed into the curve with equation $y = h(x)$ by the following sequence of transformations: Translation of $\begin{pmatrix}2\\3\end{pmatrix}$; Stretch in the $y$-direction by scale factor $k$; Reflection in the $x$-axis. Find an expression for $h(x)$ in terms of $x$ and $k$.
(c)[3]
It is given that the range of $h$ satisfies $h(x) \leq 15$. Find the values of $c$ and $k$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Identify the inverse function $g^{-1}(x)=\frac12(x-k)$” …