стр. 7 |

dependent variable. or in economic applications. the endogenous variable.

Polynomials

/;(I) = ixвЂ™, f?(i) = Y-. and f;(r) = IO.rвЂ™. (вЂ˜1

where we write the monomial terms of a polynomial in order of decreasing degree.

For any polynomial, the highest degree of any monomial that appears in it is called

degree

the of the polynomial. For example, the degree of the above polynomial h

is 7.

rational functions;

There are more complex types of functions: which are

ratios of polynomials, like

x5 + 7x

2+1 % I

and y =

4вЂ˜=

Y=- вЂ™ -1 y = xx + 3x + 2вЂ™

x-1 5

exponential functions, in which the variable x appears as an exponent, like

y = l(r; trigonometric functions, like y = sinx and y = cosx; and so on.

Graphs

Usually, the essential information about a function is contained in its graph. The

graph of a function of one variable consists of all points in the Cartesian plane

whose coordinates (1, y) satisfy the equation y = f(.x). In Figure 2.2 below, the

graphs of the five functions mentioned above are drawn.

Increasing and Decreasing Functions

The basic geometric properties of a function arc whcthcr it is increasing or de-

creasing and the location of its local and global minima and maxima. A function is

increasing if its graph mopes upward from left to right. More prcciscly. a function

f is increasing if

I, b xz implies that f(x,) >I

The functions in the first two graphs of Figure 2.2 are increasing functions. A

decreasing

fimction is if its graph moves downward from left to right. i.e.. if

The fourth function in Figure 2.2. h(x) = -˜r7. is a dccrcasing funclion.

The places where a function changes from increasing to dccrcasing and vice

f

versa are also important. If a function changes from decreasing to increasing at

f(.q,)). as

.x1. the graph of / turns upward around the point (xi,, in Figure 2.3. This

implies that the graph of /вЂ™ lies abovc the point (x0, f(x,,)) around that point. Such

a point (.r,,. f(x,,)) is called a local or relative minimum of the function fвЂ™. If the

f newr lies f(x,,)); f(x) 2 f(q) for all

graph of a function below (xi,. i.c.. if x,

f&)) a global or absolute minimum

then (x,), is called off. The point (0. 0) is a

f,(l)

global minimum of = 3.xвЂ™ in Figure 2.2.

L2.11 FUNCTIONS ON RвЂ™ 13

++jL y

Figure

The graphs of f(i) = Y + I, g(x) = 2x f,(x) = 3xвЂ™. f?(x) -˜ xi, and

f;(x) = IOX˜. 2.2

Figure

f 2.3

Function has a mbrimum arx,,.

Similarly, if function g changes from increasing to decreasing at z,l, the graph

of fi cups downward at (y,, g(q)) as in Figure 2.4, and (q, g(q)) is called a local or

relative maximum of g; analytically, g(x) 5 g(q) for all x neat q. If g(x) I g(q)

for all x, then (z,,, ,&)) is a glubal or absolute maximum of g. The function

fi = -1UxвЂ™ in Figure 2.2 has a local and a global maximum at (0, 0).

Figure

2.4

Domain

f,

Some functions ale detined only on proper subsets of RвЂ™. Given a function the

set /(I) is is domain off.

of numbersx at which dcfincd called the For each ofthc

RвЂ™.

five functions in Figure 2.2. the domain is all of Howcvcr˜ sincc division by

LCIO is undefined. the rational function f(x) = I /I is not detincd at x = 0. Since

RвЂ™

it is defined evcrywhcrc clsc. its domain is {Cl). There are twвЂќ reasoвЂќs why

the domain of a function might hc rcstrictcd: mathematics-based and application-

hased. The most common mathematical reasons for restricting the domain arc that

one cannot divide by zero and one cannot take the square root (or the logarithm)

of a negative number. For cxamplc. the domain of the function h, (x) = I/(xвЂ™ I)

is all I except { I, + I}; and the domain of the function /IT(X) = 9.r 7 is all

I 2 7.

The domain of a function may ;ilso he rcstrictcd by the application in which the

function uiscs. Fur example. if C(x) is the cost of producing I CWS; s is naturalI!

a positive integer. The domain of C would hc the set of posilivc integers. If WLвЂ™

rcdcfmc the cost funcCon w that I-вЂ˜(*) is the cost of producing .I ˜OIZ.Y of cars. the

F is naturally the set of nonnegative real numhcrs:

domain of

The nonnegative half-line R. is a cmnmon domain for functions which arise in

applications.

12.1 I FUNCTIONS ON RвЂ™ 15

Notation If the domain of the real-valued function y = f(x) is the setD C RвЂ™,

either for mathematics-based or application-based reasons, we write

f: D - RвЂ™.

Interval Notation

Speaking of subsets of the line, letвЂ™s review the standard notation for intervals in

RвЂ™. Given two real numbers a and b, the set of all numbers between a and b is

called an interval. If the endpoints a and b are excluded, the interval is called an

open interval and written as

(a, b) - {x E RвЂ™ : a < x < b}

If both endpoints are included in the interval, the interval is called a closed interval

and written as

[a, b] = {x E RвЂ™ : a 5 x 5 b}

стр. 7 |