Functions

A function $f$ is a rule

$$f:S \to T$$

that assigns to each element $s \in S$ a value $f(s) \in T$. The set $S$is the domain of the function $f$; the set $T$, its target. The range of $f$ is the set

$$\{ t \in T; \ t = f(s) \mbox{ for some } s \in S \} \subset T .$$

If $t \in T$, then the preimage of $t$ is defined by

$$f^{-1}(t) = \{ s \in S; f(s) = t \} .$$

Similarly, for a subset $U \subset T$,

$$f^{-1}(U) = \{ s \in S; \ f(s) = t \mbox{ for some } t \in U \} ,$$

and for a subset $U \subset S$,

$$f(U) = \{ t \in T; \ f(s) = t \mbox{ for some } s \in U \} .$$

Note that

$$f^{-1}(T) = S, \ f(S) \subset T \mbox{ and } f(\emptyset) = f^{-1}(\emptyset) = \emptyset ,$$

$$f \left (f^{-1}(U) \right ) \subset U \mbox{ but } f ^{-1}( f(R)) \supset R$$

(both inclusions may be proper). If

$$f:S \to T \mbox{ and } g:T \to U$$

are functions, the composite function

$$f \circ g :S \to U$$

is defined by the formula

$$(f \circ g) (s) = f(g(s))$$

(note that the parentheses $()$ are used with two different meanings in the above equation). If the target space $T$ has more structure, for example, if it is $\mbox{${\Bbb Z}$ }$, $\mbox{${\Bbb Q}$ }$, $\mbox{${\Bbb R}$ }$ or $\mbox{${\Bbb C}$ }$, then we can define other operations on functions. If, in these cases,

$$f:S \to T \mbox{ and } g:S \to T$$

are functions then one defines the functions $f+g$ and $fg$ by the rules

$$(f+g)(s) = f(s) + g(s) \mbox{ and } (fg)(s) = f(s)g(s) .$$

We would also like to define $\frac{f}{g}$; this is possible if the target space allows division as an operation (thus in case $T = \mbox{${\Bbb Q}$ }$, $\mbox{${\Bbb R}$ }$ or $\mbox{${\Bbb C}$ }$), but we must be careful not to divide by $0$. The domain of definition for $\frac{f}{g}$is thus $S - g^{-1}(0)$.