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Many functions, though not all by any means, are defined by a single equation:
(How does this last equation define a function?)
(How does this equation determine a function?)
There are various types of functions, and they can be combined in a variety of ways to produce other functions.It is necessary therefore to spend a fair amount of time at the beginning of this chapter to present these definitions.
If $f$ and $g$ are two complex-valued functions with the same domain $S,$ i.e., $f:S\to C$ and $g:S\to C,$ and if $c$ is a complex number, we define $f+g,\phantom{\rule{3.33333pt}{0ex}}fg,\phantom{\rule{3.33333pt}{0ex}}f/g$ (if $g\left(x\right)$ is never 0), and $cf$ by the familiar formulas:
and
If $f$ and $g$ are real-valued functions, we define functions $max(f,g)$ and $min(f,g)$ by
(the maximum of the numbers $f\left(x\right)$ and $g\left(x\right)$ ), and
(the minimum of the two numbers $f\left(x\right)$ and $g\left(x\right)$ ).
If $f$ is either a real-valued or a complex-valued function on a domain $S,$ then we say that $f$ is bounded if there exists a positive number $M$ such that $\left|f\right(x\left)\right|\le M$ for all $x\in S.$
There are two special types of functions of a real or complex variable, the even functions and the odd functions.In fact, every function that is defined on all of $R$ or $C$ (or, more generally, any function whose domain $S$ equals $-S$ ) can be written uniquely as a sum of an even part and an odd part.This decomposition of a general function into simpler parts is frequently helpful.
A function $f$ whose domain $S$ equals $-S,$ is called an even function if $f(-z)=f\left(z\right)$ for all $z$ in its domain. It is called an odd function if $f(-z)=-f\left(z\right)$ for all $z$ in its domain.
We next give the definition for perhaps the most familiar kinds of functions.
A nonzero polynomial or polynomial function is a complex-valued function of a complex variable, $p:C\to C,$ that is defined by a formula of the form
where the ${a}_{k}$ 's are complex numbers and ${a}_{n}\ne 0.$ The integer $n$ is called the degree of the polynomial $p$ and is denoted by $\text{deg}\left(p\right).$ The numbers ${a}_{0},{a}_{1},...,{a}_{n}$ are called the coefficients of the polynomial.The domain of a polynomial function is all of $C;$ i.e., $p\left(z\right)$ is defined for every complex number $z.$
For technical reasons of consistency, the identically 0 function is called the zero polynomial . All of its coefficients are 0 and its degree is defined to be $-\infty .$
A rational function is a function $r$ that is given by an equation of the form $r\left(z\right)=p\left(z\right)/q\left(z\right),$ where $q$ is a nonzero polynomial and $p$ is a (possibly zero) polynomial. The domain of a rational functionis the set $S$ of all $z\in C$ for which $q\left(z\right)\ne 0,$ i.e., for which $r\left(z\right)$ is defined.
Two other kinds of functions that are simple and important are step functions and polygonal functions.
Let $[a,b]$ be a closed bounded interval of real numbers. By a partition of $[a,b]$ we mean a finite set $P=\{{x}_{0}<{x}_{1}<...<{x}_{n}\}$ of $n+1$ points, where ${x}_{0}=a$ and ${x}_{n}=b.$
The $n$ intervals $\left\{[{x}_{i-1},{x}_{i}]\right\},$ for $1\le i\le n,$ are called the closed subintervals of the partition $P,$ and the $n$ intervals $\left\{({x}_{i-1},{x}_{i})\right\}$ are called the open subintervals of $P.$
We write $\xe2\u02c6\yen P\xe2\u02c6\yen $ for the maximum of the numbers (lengths of the subintervals) $\{{x}_{i}-{x}_{i-1}\},$ and call the number $\xe2\u02c6\yen P\xe2\u02c6\yen $ the mesh size of the partition $P.$
A function $h:[a,b]\to C$ is called a step function if there exists a partition $P=\{{x}_{0}<{x}_{1}<...<{x}_{n}\}$ of $[a,b]$ and $n$ numbers $\{{a}_{1},{a}_{2},...,{a}_{n}\}$ such that $h\left(x\right)={a}_{i}$ if ${x}_{i-1}<x<{x}_{i}.$ That is, $h$ is a step function if it is a constant function on each of the (open) subintervals $({x}_{i-1},{x}_{i})$ determined by a partition $P.$ Note that the values of a step function at the points $\left\{{x}_{i}\right\}$ of the partition are not restricted in any way.
A function $l:[a,b]\to R$ is called a polygonal function , or a piecewise linear function , if there exists a partition $P=\{{x}_{0}<{x}_{1}<...<{x}_{n}\}$ of $[a,b]$ and $n+1$ numbers $\{{y}_{0},{y}_{1},...,{y}_{n}\}$ such that for each $x\in [{x}_{i-1},{x}_{i}],$ $l\left(x\right)$ is given by the linear equation
where ${m}_{i}=({y}_{i}-{y}_{{i}_{1}})/({x}_{i}-{x}_{i-1}).$ That is, $l$ is a polygonal function if it is a linear function on each of the closed subintervals $[{x}_{i-1},{x}_{i}]$ determined by a partition $P.$ Note that the values of a piecewise linear function at the points $\left\{{x}_{i}\right\}$ of the partition $P$ are the same, whether we think of ${x}_{i}$ in the interval $[{x}_{i-1},{x}_{i}]$ or $[{x}_{i},{x}_{i+1}].$ (Check the two formulas for $l\left({x}_{i}\right).$ )
The graph of a piecewise linear function is the polygonal line joining the $n+1$ points $\left\{({x}_{i},{y}_{i})\right\}.$
There is a natural generalization of the notion of a step function that works for any domain $S,$ e.g., a rectangle in the plane $C.$ Thus, if $S$ is a set, we define a partition of $S$ to be a finite collection $\{{E}_{1},{E}_{2},...,{E}_{n}\}$ of subsets of $S$ for which
Then, a step function on $S$ would be a function $h$ that is constant on each subset ${E}_{i}.$ We will encounter an even more elaborate generalized notion of a step function in Chapter V, but for now we will restrict our attention to step functions defined on intervals $[a,b].$
The set of polynomials and the set of step functions are both closed under additionand multiplication, and the set of rational functions is closed under addition,multiplication, and division.
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