CK12 Calculus - Single Variable

Let’s startwith the notationwe use to describefunctions.Considerthe exampleof the linearfunction
We couldalso describethe functionusingthe symbol and read as “ of ” to indicate
the -valueof the functionfor a particular -value.In particular, for this functionwe wouldwrite
and indicatethe valueof the functionat a particularvalue,say as and find
its valueas follows: This statementcorrespondsto the solution as a point
on the graphof the function.It is read,“ of is .”
We can now beginto discussthe propertiesof functions,startingwith the domain and the range of a
function.The domainrefersto the set of -valuesthat are inputsin the function,whiletherangerefers
to the set of -valuesthat the functiontakeson. Recallour examplesof functions:

LinearFunction

QuadraticFunction

PolynomialFunction

We first note that we couldinsertany real numberfor an -valueand a well-defined -valuewouldcome
out. Henceeachfunctionhas the set of all real numbersas a domainand we indicatethis in intervalform
as. Likewisewe see that our graphscouldextendup in a positivedirectionand downin a
negativedirectionwithoutend in eitherdirection.Hencewe see that the set of -values,or the range,is
the set of all real numbers
Example2:
Determinethe domainand rangeof the function.

Solution:We note that the conditionfor each -valueis a fractionthat includesan term in the denomi-
nator. In decidingwhat set of -valueswe can use, we needto excludethosevaluesthat makethe denom-
inatorequalto Why?(Answer:divisionby is not definedfor real numbers.)Hencethe set of all
permissible -values,is all real numbersexceptfor the numbers whichyielddivisionby zero.
So on our graphwe will not see any pointsthat correspondto these -values.It is moredifficult to find the
range,so let’s find it by usingthe graphingcalculatorto producethe graph.

Fromthe graph,we see that every valuein (or "All real numbers")is represented;hence
the rangeof the functionis This is becausea fractionwith a non-zeronumerator
neverequalszero.
EightBasicFunctions