Replacing
whichis the inverseof the originalfunction.
Graphsof InverseFunctions
Whatis the relationshipbetweenthe graphsof and? If the point is on the graphof
then from the definitionof the inverse,the point is on the graphof In otherwords,whenwe
reversethe coordinatesof a pointon the graphof we automaticallyget a pointon the graphof
We concludethat and are reflectionsof one anotheraboutthe line That is, each
is a mirrorimageof the otheraboutthe line The figurebelowshowsan exampleof and,
whenthe domainis restricted,its inverse and how they are reflectedabout.
It is importantto note that for the function to havean inverse,we mustrestrictits domainto
sincethat is the domainin whichthe functionis increasing.
Continuityand Differentiabilityof InverseFunctions
Sincethe graphof a one-to-onefunctionand its inverseare reflectionsof one anotheraboutthe line
it wouldbe safe to say that if the function has no breaks(no discontinuities)then will not havebreaks
either. This impliesthat if is continuouson the domain then its inverse is continuouson the
range of For example,if , then its domainis and its rangeis This means