082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

60 Chapter 2 Engineering functions
2.3.3 One-to-many
Some rules relating input to output are not functions. Consider the rule: ‘take plus or
minus the square rootof the input’, that is
x→± √ x
Now, for example, if 4 is the input, the output is ± √ 4 which can be 2 or −2. Thus a
single input has produced more than one output. The rule is said to be one-to-many,
meaning that one input has produced many outputs. Rules with this property are not
functions. For a ruletobe a function there mustbe a single output forany given input.
Bydefiningarulemorespecifically,itmaybecomeafunction.Forexample,consider
therule:‘takethepositivesquarerootoftheinput’.Thisruleisafunctionbecausethere
is a single output for a given input. Note that the domain of this function is [0,∞) and
the range isalso[0,∞).
2.3.4 Many-to-oneandone-to-onefunctions
Consideragainthefunction f (x) =x 2 giveninExample2.4.Theinputs2and −2both
producethesameoutput,4,andthefunctionissaidtobemany-to-one.Thismeansthat
many inputs produce the same output. A many-to-one function can be recognized from
itsgraph.Ifahorizontallineintersectsthegraphinmorethanoneplace,thefunctionis
many-to-one. Figure 2.7 illustrates a many-to-one function,g(x). The inputsx 1
,x 2
,x 3
andx 4
all produce the same output.
Afunctionisone-to-oneifdifferentinputsalwaysproducedifferentoutputs.Ahorizontallinewillintersectthegraphofaone-to-onefunctioninonlyoneplace.Figure2.8
illustratesaone-to-one function,h(x).
Bothone-to-onefunctionsandmany-to-onefunctionsaresupportedintechnicalcomputinglanguages.Forexample,inMATLAB
® thefunction f (x) =x 2 canbedefinedby
using the command:
f = @(x) x^2;
Itis now possible totype:
f(3)
or
f(-3)
g(x)
h(x)
x 1 x 2 x 3 x 4 x
Figure2.7
Theinputsx 1 ,x 2 ,x 3 andx 4 allproduce the same
output,thereforeg(x) isamany-to-one function.
Figure2.8
Eachinput producesadifferent outputandso
h(x)is aone-to-onefunction.
x