Of these,the circlehas a qualitythat the othergraphsdo not share.Do you knowwhatit is?
Solution:
The circle’s graphincludespointswherea particular -valuehas two pointsassociatedwith it; for example,
the points and are both solutionsto the equation For eachof the other
relationships,a particular -valuehas exactlyone -valueassociatedwith it.)
The relationshipsthat satisfythe conditionthat for each -valuethereis a unique -valueare called
functions. Notethat we couldhavedeterminedwhetherthe relationshipsatisfiedthis conditionby a
graphicaltest, the verticalline test. Recallthe relationshipsof the circle,whichis not a function.Let’s compare
it with the parabola,whichis a function.
If we drawverticallines throughthe graphsas indicated,we see that the conditionof a particular -value
havingexactlyone -valueassociatedwith it is equivalentto havingat mostone pointof intersectionwith
any verticalline. The lines on the circleintersectthe graphin morethan one point,whilethe lines drawnon
the parabolaintersectthe graphin exactlyone point.So this verticalline test is a quickand easyway to
checkwhetheror not a graphdescribesa function.
We wantto examinepropertiesof functionssuchas functionnotation,their domainand range(the sets of
and valuesthat definethe function),graphsketchingtechniques,how we can combinefunctionsto
get new functions,and also surveysomeof the basicfunctionsthat we will deal with throughoutthe rest of
this book.