3.1 Functions  (Page 2/2)

Many functions, though not all by any means, are defined by a single equation:

$y=3x-7,$
$y={\left({x}^{2}+x+1\right)}^{2/3},$
${x}^{2}+{y}^{2}=4,$

(How does this last equation define a function?)

${\left(1-{x}^{7}{y}^{11}\right)}^{2/3}={\left(x/\left(1-y\right)\right)}^{8/17}.$

(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:

$\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right),$
$\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right),$
$\left(f/g\right)\left(x\right)=f\left(x\right)/g\left(x\right),$

and

$\left(cf\right)\left(x\right)=cf\left(x\right).$

If $f$ and $g$ are real-valued functions, we define functions $max\left(f,g\right)$ and $min\left(f,g\right)$ by

$\left[max\left(f,g\right)\right]\left(x\right)=max\left(f\left(x\right),g\left(x\right)\right)$

(the maximum of the numbers $f\left(x\right)$ and $g\left(x\right)$ ), and

$\left[min\left(f,g\right)\right]\left(x\right)=min\left(f\left(x\right),g\left(x\right)\right),$

(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 $|f\left(x\right)|\le M$ 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 $f\left(-z\right)=f\left(z\right)$ for all $z$ in its domain. It is called an odd function if $f\left(-z\right)=-f\left(z\right)$ 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

$p\left(z\right)=\sum _{k=0}^{n}{a}_{k}{z}^{k}={a}_{0}+{a}_{1}z+{a}_{2}{z}^{2}+...+{a}_{n}{z}^{n},$

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 $\left[a,b\right]$ be a closed bounded interval of real numbers. By a partition of $\left[a,b\right]$ we mean a finite set $P=\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ of $n+1$ points, where ${x}_{0}=a$ and ${x}_{n}=b.$

The $n$ intervals $\left\{\left[{x}_{i-1},{x}_{i}\right]\right\},$ for $1\le i\le n,$ are called the closed subintervals of the partition $P,$ and the $n$ intervals $\left\{\left({x}_{i-1},{x}_{i}\right)\right\}$ are called the open subintervals of $P.$

We write $\parallel P\parallel$ for the maximum of the numbers (lengths of the subintervals) $\left\{{x}_{i}-{x}_{i-1}\right\},$ and call the number $\parallel P\parallel$ the mesh size of the partition $P.$

A function $h:\left[a,b\right]\to C$ is called a step function if there exists a partition $P=\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ of $\left[a,b\right]$ and $n$ numbers $\left\{{a}_{1},{a}_{2},...,{a}_{n}\right\}$ such that $h\left(x\right)={a}_{i}$ if ${x}_{i-1} That is, $h$ is a step function if it is a constant function on each of the (open) subintervals $\left({x}_{i-1},{x}_{i}\right)$ 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 $l:\left[a,b\right]\to R$ is called a polygonal function , or a piecewise linear function , if there exists a partition $P=\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ of $\left[a,b\right]$ and $n+1$ numbers $\left\{{y}_{0},{y}_{1},...,{y}_{n}\right\}$ such that for each $x\in \left[{x}_{i-1},{x}_{i}\right],$ $l\left(x\right)$ is given by the linear equation

$l\left(x\right)={y}_{i-1}+{m}_{i}\left(x-{x}_{i-1}\right),$

where ${m}_{i}=\left({y}_{i}-{y}_{{i}_{1}}\right)/\left({x}_{i}-{x}_{i-1}\right).$ That is, $l$ is a polygonal function if it is a linear function on each of the closed subintervals $\left[{x}_{i-1},{x}_{i}\right]$ 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 $\left[{x}_{i-1},{x}_{i}\right]$ or $\left[{x}_{i},{x}_{i+1}\right].$ (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 $\left\{\left({x}_{i},{y}_{i}\right)\right\}.$

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 $\left\{{E}_{1},{E}_{2},...,{E}_{n}\right\}$ of subsets of $S$ for which

1.   ${\cup }_{i=1}^{n}{E}_{i}=S,$ and
2.   ${E}_{i}\cap {E}_{j}=\varnothing$ if $i\ne j.$

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 $\left[a,b\right].$

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.

1. Prove that the sum and product of two polynomials is again a polynomial.Show that $\text{deg}\left(p+q\right)\le max\left(\text{deg}\left(p\right),\text{deg}\left(q\right)\right)$ and $\text{deg}\left(pq\right)=\text{deg}\left(p\right)+\text{deg}\left(q\right).$ Show that a constant function is a polynomial, and that the degree of a nonzero constant function is 0.
2. Show that the set of step functions is closed under addition and multiplication. Show also that the maximum and minimum of twostep functions is again a step function. (Be careful to note that different step functions may be determined bydifferent partitions. For instance, a partition determining the sum of two step functionsmay be different from the partitions determining the two individual step functions.) Note, in fact, that a step function can be determined by infinitely many different partitions.Prove that the sum, the maximum, and the minimum of two piecewise linear functions is again a piecewise linear function.Show by example that the product of two piecewise linear functions need not be piecewise linear.
3. Prove that the sum, product, and quotient of two rational functions is again a rational function.
4. Prove the Root Theorem: If $p\left(z\right)={\sum }_{k=0}^{n}{a}_{k}{z}^{k}$ is a nonzero polynomial of degree $n,$ and if $c$ is a complex number for which $p\left(c\right)=0,$ then there exists a nonzero polynomial $q\left(z\right)={\sum }_{j=0}^{n-1}{b}_{j}{z}^{j}$ of degree $n-1$ such that $p\left(z\right)=\left(z-c\right)q\left(z\right)$ for all $z.$ That is, if $c$ is a “root” of $p,$ then $z-c$ is a factor of $p.$ Show also that the leading coefficient ${b}_{n-1}$ of $q$ equals the leading coefficient ${a}_{n}$ of $p.$ HINT: Write
$p\left(z\right)=p\left(z\right)-p\left(c\right)=\sum _{k=0}^{n}{a}_{k}\left({z}^{k}-{c}^{k}\right)=....$
5. Let $f$ be a function whose domain $S$ equals $-S.$ Define functions ${f}_{e}$ and ${f}_{o}$ by the formulas
${f}_{e}\left(z\right)=\frac{f\left(z\right)+f\left(-z\right)}{2}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{f}_{o}\left(z\right)=\frac{f\left(z\right)-f\left(-z\right)}{2}.$
Show that ${f}_{e}$ is an even function, that ${f}_{o}$ is an odd function, and that $f={f}_{e}+{f}_{o}.$ Show also that, if $f=g+h,$ where $g$ is an even function and $h$ is an odd function, then $g={f}_{e}$ and $h={f}_{o}.$ That is, there is only one way to write $f$ as the sum of an even function and an odd function.
6. Use part (e) to show that a polynomial $p$ is an even function if and only if its only nonzero coefficients are even ones, i.e., the ${a}_{2k}$ 's. Show also that a polynomial is an odd function if and only if itsonly nonzero coefficients are odd ones, i.e., the ${a}_{2k+1}$ 's.
7. Suppose $p\left(z\right)={\sum }_{k=0}^{n}{a}_{2k}{z}^{2k}$ is a polynomial that is an even function. Show that
$p\left(iz\right)=\sum _{k=0}^{n}{\left(-1\right)}^{k}{a}_{2k}{z}^{2k}={p}^{a}\left(z\right),$
where ${p}^{a}$ is the polynomial obtained from $p$ by alternating the signs of its nonzero coefficients.
8. If $q\left(z\right)={\sum }_{k=0}^{n}{a}_{2k+1}{z}^{2k+1}$ is a polynomial that is an odd function, show that
$q\left(iz\right)=i\sum _{k=0}^{n}{\left(-1\right)}^{k}{a}_{2k+1}{z}^{2k+1}=i{q}^{a}\left(z\right),$
where again ${q}^{a}$ is the polynomial obtained from $q$ by alternating the signs of its nonzero coefficients.
9. If $p$ is any polynomial, show that
$p\left(iz\right)={p}_{e}\left(iz\right)+{p}_{o}\left(iz\right)={p}_{e}^{a}\left(z\right)+i{p}_{o}^{a}\left(z\right),$
and hence that ${p}_{e}\left(iz\right)={p}_{e}^{a}\left(z\right)$ and ${p}_{o}\left(iz\right)=i{p}_{o}^{a}\left(z\right).$

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