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\frame{\frametitle{Getting precise: Notion of a function.}
  \[f:A\to B,\quad x\mapsto f(x)\]
  means that $f$ is a function with domain $\mathcal{D}(f)=A$ and with
  values in $B$, i.e. for all $x\in A$ the image $f(x)$ belongs to
  $B$.\pause
  \begin{block}{Domain of a function}
    $\mathcal{D}(f)$ is a subset of $\mathds{R}$, which consists at most of all
    those $x\in\mathds{R}$ such that the mapping rule is well-defined.
  \end{block}
  Example: $f(x)=\tfrac{1}{x}$, then $0\not\in\mathcal{D}(f)$.\\ \pause
  \begin{block}{Range of a function}
    $\mathcal{R}(f)=\{f(x):x\in\mathcal{D}(f)\}$, i.e. the range of $f$ is a subset of $B$
    consisting of all images of $f$.
  \end{block}
}