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

## Question

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 $h^{-1}(x)+f^{-1}(x)=0$.