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Functions

Module by: Lawrence Baggett. E-mail the author

Summary: A module about functions. Many terms, such as graph, real-valued, complex-valued, imaginary, bounded, even, odd, and others are defined. An exercise at the end involves some practice by proving statements and theorems related to these definitions of functions.

Definition 1:

Let SS and TT be sets. A function from SS into TT (notation f:STf:ST) is a rule that assigns to each element xx in SS a unique element denoted by f(x)f(x) in T.T.

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

If f:STf:ST is a function, then SS is called the domain of f,f, and the set TT is called the codomain of f.f. The range or image of ff is the set of all elements yy in the codomain TT for which there exists an xx in the domain SS such that y=f(x).y=f(x). We denote the range by f(S).f(S). The codomain is the set of all potential outputs, while the range is the set of actual outputs.

Suppose ff is a function from a set SS into a set T.T. If AS,AS, we write f(A)f(A) for the subset of TT containing all the elements tTtT for which there exists an sAsA such that t=f(s).t=f(s). We call f(A)f(A) the image of AA under f.f. Similarly, if BT,BT, we write f-1(B)f-1(B) for the subset of SS containing all the elements sSsS such that f(s)B,f(s)B, and we call the set f-1(B)f-1(B) the inverse image or preimage of B.B. The symbol f-1(B)f-1(B) is a little confusing, since it could be misinterpreted as the image of the set BB under a function called f-1.f-1. We will discuss inverse functions later on, but this notation is not meant to imply that the function ff has an inverse.

If f:ST,f:ST, then the graph of ff is the subset GG of the Cartesian product S×TS×T consisting of all the pairs of the form (x,f(x)).(x,f(x)).

If f:SRf:SR is a function, then we call ff a real-valued function, and if f:SC,f:SC, then we call ff a complex-valued function. If f:SCf:SC is a complex-valued function, then for each xSxS the complex number f(x)f(x) can be written as u(x)+iv(x),u(x)+iv(x), where u(x)u(x) and v(x)v(x) are the real and imaginary parts of the complex number f(x).f(x). The two real-valued functions u:SRu:SR and v:SRv:SR are called respectively the real and imaginary parts of the complex-valued function f.f.

If f:STf:ST and SR,SR, then ff is called a function of a real variable, and if SC,SC, then ff is called a function of a complex variable.

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

The function f:STf:ST is called one-to-one if f(x1)=f(x2)f(x1)=f(x2) implies that x1=x2.x1=x2.

The domain of ff is the set of xx's for which f(x)f(x) is defined. If we are given a function f:ST,f:ST, we are free to regard ff as having a smaller domain, i.e., a subset S'S' of S.S. Although this restricted function is in reality a different function, we usually continue to call it by the same name f.f. Enlarging the domain of a function, in some consistent manner, is often impossible, but is nevertheless frequently of great importance. The codomain of ff 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:SRf:SR 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 RR or of C.C. Frequently, we will indicate what kind of variable we are thinking of by denoting real variables with the letter xx and complex variables with the letter z.z. Be careful about this, for this distinction is not always made.

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

y = 3 x - 7 , y = 3 x - 7 ,
(1)
y = ( x 2 + x + 1 ) 2 / 3 , y = ( x 2 + x + 1 ) 2 / 3 ,
(2)
x 2 + y 2 = 4 , x 2 + y 2 = 4 ,
(3)

(How does this last equation define a function?)

( 1 - x 7 y 11 ) 2 / 3 = ( x / ( 1 - y ) ) 8 / 17 . ( 1 - x 7 y 11 ) 2 / 3 = ( x / ( 1 - y ) ) 8 / 17 .
(4)

(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.

Definition 2:

If ff and gg are two complex-valued functions with the same domain S,S, i.e., f:SCf:SC and g:SC,g:SC, and if cc is a complex number, we define f+g,fg,f/gf+g,fg,f/g (if g(x)g(x) is never 0), and cfcf by the familiar formulas:

( f + g ) ( x ) = f ( x ) + g ( x ) , ( f + g ) ( x ) = f ( x ) + g ( x ) ,
(5)
( f g ) ( x ) = f ( x ) g ( x ) , ( f g ) ( x ) = f ( x ) g ( x ) ,
(6)
( f / g ) ( x ) = f ( x ) / g ( x ) , ( f / g ) ( x ) = f ( x ) / g ( x ) ,
(7)

and

( c f ) ( x ) = c f ( x ) . ( c f ) ( x ) = c f ( x ) .
(8)

If ff and gg are real-valued functions, we define functions max(f,g)max(f,g) and min(f,g)min(f,g) by

[ max ( f , g ) ] ( x ) = max ( f ( x ) , g ( x ) ) [ max ( f , g ) ] ( x ) = max ( f ( x ) , g ( x ) )
(9)

(the maximum of the numbers f(x)f(x) and g(x)g(x)), and

[ min ( f , g ) ] ( x ) = min ( f ( x ) , g ( x ) ) , [ min ( f , g ) ] ( x ) = min ( f ( x ) , g ( x ) ) ,
(10)

(the minimum of the two numbers f(x)f(x) and g(x)g(x)).

If ff is either a real-valued or a complex-valued function on a domain S,S, then we say that ff is bounded if there exists a positive number MM such that |f(x)|M|f(x)|M for all xS.xS.

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 RR or CC (or, more generally, any function whose domain SS equals -S-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.

Definition 3:

A function ff whose domain SS equals -S,-S, is called an even function if f(-z)=f(z)f(-z)=f(z) for all zz in its domain. It is called an odd function if f(-z)=-f(z)f(-z)=-f(z) for all zz in its domain.

We next give the definition for perhaps the most familiar kinds of functions.

Definition 4:

A nonzero polynomial or polynomial function is a complex-valued function of a complex variable, p:CC,p:CC, that is defined by a formula of the form

p ( z ) = k = 0 n a k z k = a 0 + a 1 z + a 2 z 2 + ... + a n z n , p ( z ) = k = 0 n a k z k = a 0 + a 1 z + a 2 z 2 + ... + a n z n ,
(11)

where the akak's are complex numbers and an0.an0. The integer nn is called the degree of the polynomial pp and is denoted by deg(p).deg(p). The numbers a0,a1,...,ana0,a1,...,an are called the coefficients of the polynomial. The domain of a polynomial function is all of C;C; i.e., p(z)p(z) is defined for every complex number z.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 -.-.

A rational function is a function rr that is given by an equation of the form r(z)=p(z)/q(z),r(z)=p(z)/q(z), where qq is a nonzero polynomial and pp is a (possibly zero) polynomial. The domain of a rational function is the set SS of all zCzC for which q(z)0,q(z)0, i.e., for which r(z)r(z) is defined.

Two other kinds of functions that are simple and important are step functions and polygonal functions.

Definition 5:

Let [a,b][a,b] be a closed bounded interval of real numbers. By a partition of [a,b][a,b] we mean a finite set P={x0<x1<...<xn}P={x0<x1<...<xn} of n+1n+1 points, where x0=ax0=a and xn=b.xn=b.

The nn intervals {[xi-1,xi]},{[xi-1,xi]}, for 1in,1in, are called the closed subintervals of the partition P,P, and the nn intervals {(xi-1,xi)}{(xi-1,xi)} are called the open subintervals of P.P.

We write PP for the maximum of the numbers (lengths of the subintervals) {xi-xi-1},{xi-xi-1}, and call the number PP the mesh size of the partition P.P.

A function h:[a,b]Ch:[a,b]C is called a step function if there exists a partition P={x0<x1<...<xn}P={x0<x1<...<xn} of [a,b][a,b] and nn numbers {a1,a2,...,an}{a1,a2,...,an} such that h(x)=aih(x)=ai if xi-1<x<xi.xi-1<x<xi. That is, hh is a step function if it is a constant function on each of the (open) subintervals (xi-1,xi)(xi-1,xi) determined by a partition P.P. Note that the values of a step function at the points {xi}{xi} of the partition are not restricted in any way.

A function l:[a,b]Rl:[a,b]R is called a polygonal function, or a piecewise linear function, if there exists a partition P={x0<x1<...<xn}P={x0<x1<...<xn} of [a,b][a,b] and n+1n+1 numbers {y0,y1,...,yn}{y0,y1,...,yn} such that for each x[xi-1,xi],x[xi-1,xi],l(x)l(x) is given by the linear equation

l ( x ) = y i - 1 + m i ( x - x i - 1 ) , l ( x ) = y i - 1 + m i ( x - x i - 1 ) ,
(12)

where mi=(yi-yi1)/(xi-xi-1).mi=(yi-yi1)/(xi-xi-1). That is, ll is a polygonal function if it is a linear function on each of the closed subintervals [xi-1,xi][xi-1,xi] determined by a partition P.P. Note that the values of a piecewise linear function at the points {xi}{xi} of the partition PP are the same, whether we think of xixi in the interval [xi-1,xi][xi-1,xi] or [xi,xi+1].[xi,xi+1]. (Check the two formulas for l(xi).l(xi).)

The graph of a piecewise linear function is the polygonal line joining the n+1n+1 points {(xi,yi)}.{(xi,yi)}.

There is a natural generalization of the notion of a step function that works for any domain S,S, e.g., a rectangle in the plane C.C. Thus, if SS is a set, we define a partition of SS to be a finite collection {E1,E2,...,En}{E1,E2,...,En} of subsets of SS for which

  1.  i=1nEi=S,i=1nEi=S, and
  2.  EiEj=EiEj= if ij.ij.

Then, a step function on SS would be a function hh that is constant on each subset Ei.Ei. 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].[a,b].

The set of polynomials and the set of step functions are both closed under addition and multiplication, and the set of rational functions is closed under addition, multiplication, and division.

Exercise 1

  1. Prove that the sum and product of two polynomials is again a polynomial. Show that deg(p+q)max(deg(p),deg(q))deg(p+q)max(deg(p),deg(q)) and deg(pq)=deg(p)+deg(q).deg(pq)=deg(p)+deg(q). 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 two step functions is again a step function. (Be careful to note that different step functions may be determined by different partitions. For instance, a partition determining the sum of two step functions may 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(z)=k=0nakzkp(z)=k=0nakzk is a nonzero polynomial of degree n,n, and if cc is a complex number for which p(c)=0,p(c)=0, then there exists a nonzero polynomial q(z)=j=0n-1bjzjq(z)=j=0n-1bjzj of degree n-1n-1 such that p(z)=(z-c)q(z)p(z)=(z-c)q(z) for all z.z. That is, if cc is a “root” of p,p, then z-cz-c is a factor of p.p. Show also that the leading coefficient bn-1bn-1 of qq equals the leading coefficient anan of p.p. HINT: Write
    p(z)=p(z)-p(c)=k=0nak(zk-ck)=....p(z)=p(z)-p(c)=k=0nak(zk-ck)=....
    (13)
  5. Let ff be a function whose domain SS equals -S.-S. Define functions fefe and fofo by the formulas
    fe(z)=f(z)+f(-z)2andfo(z)=f(z)-f(-z)2.fe(z)=f(z)+f(-z)2andfo(z)=f(z)-f(-z)2.
    (14)
    Show that fefe is an even function, that fofo is an odd function, and that f=fe+fo.f=fe+fo. Show also that, if f=g+h,f=g+h, where gg is an even function and hh is an odd function, then g=feg=fe and h=fo.h=fo. That is, there is only one way to write ff as the sum of an even function and an odd function.
  6. Use part (e) to show that a polynomial pp is an even function if and only if its only nonzero coefficients are even ones, i.e., the a2ka2k's. Show also that a polynomial is an odd function if and only if its only nonzero coefficients are odd ones, i.e., the a2k+1a2k+1's.
  7. Suppose p(z)=k=0na2kz2kp(z)=k=0na2kz2k is a polynomial that is an even function. Show that
    p(iz)=k=0n(-1)ka2kz2k=pa(z),p(iz)=k=0n(-1)ka2kz2k=pa(z),
    (15)
    where papa is the polynomial obtained from pp by alternating the signs of its nonzero coefficients.
  8. If q(z)=k=0na2k+1z2k+1q(z)=k=0na2k+1z2k+1 is a polynomial that is an odd function, show that
    q(iz)=ik=0n(-1)ka2k+1z2k+1=iqa(z),q(iz)=ik=0n(-1)ka2k+1z2k+1=iqa(z),
    (16)
    where again qaqa is the polynomial obtained from qq by alternating the signs of its nonzero coefficients.
  9. If pp is any polynomial, show that
    p(iz)=pe(iz)+po(iz)=pea(z)+ipoa(z),p(iz)=pe(iz)+po(iz)=pea(z)+ipoa(z),
    (17)
    and hence that pe(iz)=pea(z)pe(iz)=pea(z) and po(iz)=ipoa(z).po(iz)=ipoa(z).

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