Functions f and g are defined by
f: x \mapsto 2x + 1, \quad x \in \mathbb{R}, x > 0,
g: x \mapsto \frac{2x - 1}{x + 3}, \quad x \in \mathbb{R}, x \neq -3.
(i) Solve the equation gf(x) = x.
(ii) Express $f^{-1}(x)$ and $g^{-1}(x)$ in terms of x.
(iii) Show that the equation $g^{-1}(x) = x$ has no solutions.
(iv) Sketch in a single diagram the graphs of y = f(x) and y = f?¹(x), making clear the relationship between the graphs.
