Notational note

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 \[h(x)+f(x)=0.\]

Note that the symbol \(\mathbb{R}\) refers to the set of all real numbers, that is, all the numbers on the number line.

Similarly, we denote the integers \((...,-2,-1,0,1,2,...)\) by \(\mathbb{Z}\), and the rationals (numbers \(a/b\), where \(a\) and \(b\) are whole numbers, \(b \neq 0\)) by \(\mathbb{Q}\).

Adding a small subscript “\(+\)” after any of these symbols means we only include the positive members of the relevant set. So, for example, \(\mathbb{R}_+\) refers to the positive real numbers. Similarly, a small subscript “\(-\)” would mean include only the negative numbers in this set.

An alternative notation which is also quite common is to use a superscript “\(+\)” instead, for example \(\mathbb{R}^+\).