Lessonon Relationsand Functions,particularlyrational,polynomial,radical,and trigonometricfunctions.
Beforewe beginour workon theseexamples,it may be usefulto summarizethe kind of informationabout
functionswe now can generatebasedon our previousdiscussions.Let'ssummarizeour resultsin a table
like the one shownbecauseit providesa usefultemplatewith whichto organizeour findings.
Table Summary
f(x) Analysis
Domainand Range
Interceptsand Zeros
Asymptotesand limitsat infin-
ity
Differentiability
Intervalswheref is increasing
Intervalswheref is decreasing
Relativeextrema
Concavity
InflectionpointsExample1:AnalyzingRationalFunctions
Considerthe function
GeneralProperties:The functionappearsto havezerosat However, oncewe factorthe expres-
sion we see
.Hence,the functionhas a zeroat thereis a holein the graphat the domainis
and the -interceptis at
Asymptotesand Limitsat Infinity
Giventhe domain,we note that thereis a verticalasymptoteat To determineotherasymptotes,we
examinethe limit of as and. We have
.
Similarly, we see that. We also note that since
Hencewe havea horizontalasymptoteat
Differentiability