Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 3] FUNCTIONS AND ALGORITHMS 45

EXAMPLE 3.2

(a) Letf:A→Bbe the function defined in Example 3.1 (b). Then the graph offis as follows:

{(a, s), (b, u), (c, r), (d, s)}

(b) Consider the following three relations on the setA={ 1 , 2 , 3 }:

f={( 1 , 3 ), ( 2 , 3 ), ( 3 , 1 )},g={( 1 , 2 ), ( 3 , 1 )},h={( 1 , 3 ), ( 2 , 1 ), ( 1 , 2 ), ( 3 , 1 )}

fis a function fromAintoAsince each member ofAappears as the first coordinate in exactly one ordered

pair inf; heref( 1 )= 3 ,f( 2 )=3, andf( 3 )=1.gis not a function fromAintoAsince 2 ∈Aisnot the

first coordinate of any pair ingand sogdoes not assign any image to 2. Alsohis not a function fromAinto

Asince 1∈Aappears as the first coordinate of two distinct ordered pairs inh,( 1 , 3 )and( 1 , 2 ).Ifhis to be

a function it cannot assign both 3 and 2 to the element 1∈A.

(c) By areal polynomial function, we mean a functionf:R→Rof the form

f(x)=anxn+an− 1 xn−^1 +···+a 1 x+a 0

where theaiare real numbers. SinceRis an infinite set, it would be impossible to plot each point of the

graph. However, the graph of such a function can be approximated by first plotting some of its points and then

drawing a smooth curve through these points. The points are usually obtained from a table where various

values are assigned toxand the corresponding values off(x)are computed. Figure 3-2 illustrates this

technique using the functionf(x)=x^2 − 2 x−3.

Fig. 3-2

Composition Function

Consider functionsf:A→Bandg:B→C; that is, where the codomain offis the domain ofg. Then we
may define a new function fromAtoC, called thecompositionoffandgand writteng◦f, as follows:

(g◦f )(a)≡g(f (a))

That is, we find the image ofaunderfand then find the image off(a)underg. This definition is not really
new. If we viewfandgas relations, then this function is the same as the composition offandgas relations (see
Section 2.6) except that here we use the functional notationg◦ffor the composition offandginstead of the
notationf◦gwhich was used for relations.