Consider the functions

f : A R defined by f(x) = log₉ (x² 18x + 81),

g : B R defined by g(x) = 1 /(3 ^x 9 ), and

h : C R defined by h = g f,

where A, B and C are the maximal domains given by these definitions.

(a) Determine A and B.

(b) Find a piecewise expression for h(x), and determine C.

(c) The domain of h is now restricted to the set D = C (a,), where a R. Find the minimum value of a such that h is a one-to-one function on D.

(d) With h restricted to this domain D, determine an expression for h^-1 (x), and state the domain and range of h^-1