Problem

Functions $f$ and $g$ are defined for all real values of $x$ by

$$f(x) = 4x^2 - c \quad \text{and} \quad g(x) = 2x + k,$$

where $c$ and $k$ are positive constants. It is given that $g^{-1}(3k + 1) = c$.

(a) Show that $gf(x) = 8x^2 - k - 1$.

(b) The curve with equation $y = 8x^2 - k - 1$ is transformed to 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) The range of $h$ is given by $h(x) \leq 15$.

Find the values of $c$ and $k$.