3.7: Logarithmic function sketch and composite equation

Write down the least possible value of $a$.

Write down the range of $f$.

It is given that the equation $f(x) = f^{-1}(x)$ has two solutions. (You do not need to solve this equation.) Using your answer to part (a), sketch the graphs of $y = f(x)$ and $y = f^{-1}(x)$ on the axes below, stating the coordinates of the points where the graphs meet the axes.

It is given that $g(x) = 2x - 3$ for $x \geqslant 3$.

Find an expression for $g(g(x))$.

Hence solve the equation $fg(g(x)) = 4$ giving your answer in exact form.