Problem

The function $f$ is defined by $f(x) = 10 + 6x - x^2$ for $x \in \mathbb{R}$.

(a) By completing the square, find the range of $f$.

The function $g$ is defined by $g(x) = 4x + k$ for $x \in \mathbb{R}$ where $k$ is a constant.

(b) It is given that the graph of $y = g^{-1}f(x)$ meets the graph of $y = g(x)$ at a single point $P$.

Determine the coordinates of $P$.