3.10: Quadratic mapping inverse sketch and logarithmic composite

It is given that $f: x \mapsto (3x + 1)^{2} - 4$ for $x \geqslant a$, and that $f^{-1}$ exists.

Find the least possible value of $a$.

Using this value of $a$, write down the range of $f$.

Using this value of $a$, sketch the graphs of $y = f(x)$ and $y = f^{-1}(x)$ on the axes, stating the intercepts with the coordinate axes.

It is given that
$$g(x) = \ln(2x^{2} + 5) \quad\text{for } x \geqslant 0,$$$$h(x) = 3x - 2 \quad\text{for } x \geqslant 0.$$

Solve the equation $hg(x) = 4$ giving your answer in exact form.