Somefunctionshavesuddenjumps.Considerthe “rounding”functionthat takesa numberand roundsit to
the nearestwholenumber(roundingup if the numberis exactlybetweentwo wholenumbers).So some
valuesfor this functionare (2, 2), (1.4, 1), (3.9, 4), (5.5, 6), and (-5.5,-5).The domainof this functionis all
real numbers,but the rangeof the functionis the integers.
Anotherfunctionthat jumpscomesfrom the way taxis oftencharge.Supposea taxi costs$5.00for the first
2 milesbut then $1 for eachadditionalmile or fractionof a mile.Considerthe functionthat has the distance
traveledas the inputand the cost of the taxi ride as the output.So somevaluesfor this functionare (1, 5),
(1.9,5), (2.1,6), (10, 13). The domainof this functionis the non-negativereal numbers(sinceyou can’t
travela negativedistancein a taxi cab).The rangeof this functionis all positiveintegersgreaterthan or
equalto 5: {5, 6, 7, 8, ...}.
Example3:Statethe domainand rangeof the function
Solution:The domainand rangeof the function
For this function,we can chooseanyxvalue,except. Thereforethe domainof the functionis the
set of all real numbers,except.
The rangeis also restrictedto the non-zeroreal numbers,but for a differentreason.Becausethe numerator
of the fractionis 2, the numeratorcan neverequalzero,so the fractioncan neverequalzero.
Now that we havedefinedwhatit meansfor a relationto be a function,and we havedefineddomainand
rangeof a function,we can look at somespecificexamplesof functionsand their graphs.
Familiesof functions
The exampleswe haveseenso far haveincludedseveraldifferenttypesof functions.Fromyour previous
experienceworkingwith equationsand graphs,you may havealreadymadeconnectionsbetweenthe forms
of the equationsof functions,and whatthe graphslook like. Herewe will examineseveral“families”of
functions.A familyof functionsis a set of functionswhoseequationshavea similarform.The “parent”of
the familyis the equationin the familywith the simplestform.For example,y=x^2 is a parentto otherfunc-
tions,suchasy= 2x^2 - 5x+ 3. The tablebelowsummarizesthe key aspectsof severalfamiliesof functions.