Problem

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

$$f(x) = (3x - 2)^2 + k \quad \text{and} \quad g(x) = 5x - 1,$$

where $k$ is a constant.

(a) Given that the range of the function $gf$ is $gf(x) \geq 39$, find the value of $k$.

(b) For this value of $k$, determine the range of the function $fg$.

(c) The function $h$ is defined for all real values of $x$ and is such that $gh(x) = 35x + 19$.

Find an expression for $g^{-1}(x)$ and hence, or otherwise, find an expression for $h(x)$.