Review question

# Can we find the ranges of $f$ and $g$, and the function $f \circ g$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7607

## 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 $\mathbb{R}^+$.