A smooth curve representing an exponential function passes through the points (0,1) and (2,9). It has the horizontal asymptote y=0 and is increasing for all real x.
(i) Assuming the function is of the form y=ax, determine the value of the base a.
(ii) Sketch the graph of the function, showing the asymptote and the two given points.
(iii) Write the equation of the inverse function and state its vertical asymptote and its intercepts with the coordinate axes.