2 and with -2. Similarly, 9 is pairedwith 3 and with -3. Thereforethe relationis not a function.If we look at
the graphabove,we can see that, exceptforx= 0, thexvaluesof the relationare eachpairedwith twoy
values.Thereforethe aboverelationis not a function.
One way to quicklydeterminewhetheror not a relationis a functionis performtheverticalline test,which
meansthat you drawa verticalline throughthe graph.For example,if we drawthe linex= 4 throughthe
graphofx=y
2
, the line will intersectthe graphtwice.This meansthe relationis not a function.
Example1:Determineif the relationis a functionor not
a. (2,4),(3,9),(5,11),(5,12)
b.
Solution:
a. (2,4),(3,9),(5,11),(5,12)
This relationis not a functionbecause5 is pairedwith 11 and with 12. If you plottedthe pints,the linex= 5
wouldtouch2 pointsin the relation.
b. This relationis a functionbecauseeveryxis pairedwith only oney.
Onceyou are able to determineif a relationis a function,you shouldthen be able to statethe set ofxvalues
and the set ofyvaluesfor whicha functionis defined.
Thedomainof a functionis definedas the set of allxvaluesfor whichthe functionis defined.For example,
the domainof the functiony= 3xis the set of all real numbers,oftenwrittenas. This meansthatxcan
be any real number. Otherfunctionshaverestricteddomains.For example,the domainof the function
is the set of all real numbersgreaterthan or equalto zero.The domainof this functionis restricted
in this way becausethe squareroot of a negativenumberis not a real number. Thereforethe domainis re-
strictedto non-negativevaluesofxso that the functionvalueswill be defined.
It is ofteneasyto determinethe domainof a functionby (1) consideringwhatrestrictionstheremightbe and
(2) lookingat a graph.