Functions \(f\) and \(g\) are defined by \[\begin{align*} f:{}&x \to \log_a x,\quad&&(x \in \mathbb{R}_+, a>1),\\ g:{}&x\to \frac{1}{x},&&(x \in \mathbb{R}_+). \end{align*}\]

State the ranges of \(f\) and \(g\), and show that if \(h\) denotes the composite function \(f \circ g\), then \[h(x)+f(x)=0.\]

Explain briefly why the composite function \(g \circ f\) cannot be properly defined unless the domain is restricted to a subset of \(\mathbb{R}_+\), and state a possible subset which would be suitable.

Define fully the inverses of \(f\) and \(g\), and determine whether or not \(h^{-1}(x)+f^{-1}(x)=0\).