3.1 Functions

Let $S$ and $T$ be sets. A function from $S$ into $T$ (notation $f:S\to T$ ) is a rule that assigns to each element $x$ in $S$ a unique element denoted by $f\left(x\right)$ in $T.$

It is useful to think of a function as a mechanism or black box. We use the elements of $S$ as inputs to the function, and the outputs are elements of the set $T.$

If $f:S\to T$ is a function, then $S$ is called the domain of $f,$ and the set $T$ is called the codomain of $f.$ The range or image of $f$ is the set of all elements $y$ in the codomain $T$ for which there exists an $x$ in the domain $S$ such that $y=f\left(x\right).$ We denote the range by $f\left(S\right).$ The codomain is the set of all potential outputs, while the range is the set of actual outputs.

Suppose $f$ is a function from a set $S$ into a set $T.$ If $A\subseteq S,$ we write $f\left(A\right)$ for the subset of $T$ containing all the elements $t\in T$ for which there exists an $s\in A$ such that $t=f\left(s\right).$ We call $f\left(A\right)$ the image of $A$ under $f.$ Similarly, if $B\subseteq T,$ we write $f^{-1}\left(B\right)$ for the subset of $S$ containing all the elements $s\in S$ such that $f\left(s\right)\in B,$ and we call the set $f^{-1}\left(B\right)$ the inverse image or preimage of $B.$ The symbol $f^{-1}\left(B\right)$ is a little confusing, since it could be misinterpreted as the image of the set $B$ under a function called $f^{-1}.$ We will discuss inverse functions later on, but this notation is not meant to imply that the function $f$ has an inverse.

If $f:S\to T,$ then the graph of $f$ is the subset $G$ of the Cartesian product $S\times T$ consisting of all the pairs of the form $(x,f(x\left)\right).$

If $f:S\to R$ is a function, then we call $f$ a real-valued function, and if $f:S\to C,$ then we call $f$ a complex-valued function. If $f:S\to C$ is a complex-valued function, then for each $x\in S$ the complex number $f\left(x\right)$ can be written as $u\left(x\right)+iv\left(x\right),$ where $u\left(x\right)$ and $v\left(x\right)$ are the real and imaginary parts of the complex number $f\left(x\right).$ The two real-valued functions $u:S\to R$ and $v:S\to R$ are called respectively the real and imaginary parts of the complex-valued function $f.$

If $f:S\to T$ and $S\subseteq R,$ then $f$ is called a function of a real variable , and if $S\subseteq C,$ then $f$ is called a function of a complex variable .

If the range of $f$ equals the codomain, then $f$ is called onto .

The function $f:S\to T$ is called one-to-one if $f\left(x_1\right)=f\left(x_2\right)$ implies that $x_1=x_2.$

The domain of $f$ is the set of $x$ 's for which $f\left(x\right)$ is defined. If we are given a function $f:S\to T,$ we are free to regard $f$ as having a smaller domain, i.e., a subset $S^{\text{'}}$ of $S.$ Although this restricted function is in reality a different function, we usually continue to call it by the same name $f.$ Enlarging the domain of a function, in some consistent manner, is often impossible, but is nevertheless frequently of great importance.The codomain of $f$ is distinguished from the range of f, which is frequently a proper subset of the codomain.For example, since every real number is a complex number, any real-valued function $f:S\to R$ is also a (special kind of) complex-valued function.

We consider in this book functions either of a real variable or of complex variable. that is, the domains of functions here will be subsets either of $R$ or of $C.$ Frequently, we will indicate what kind of variable we are thinking of by denoting real variables with the letter $x$ and complex variables with the letter $z.$ Be careful about this, for this distinction is not always made.