CK12 Calculus - Single Variable

and and are inversesof eachother.

Note:In general,

Whenis a functioninvertable?It is interestingto note that if a function is alwaysincreasingor always

decreasingover its domain,then a horizontalline will cut throughthis graphat one pointonly. Then in

this caseis a one-to-onefunctionand thus has an inverse.So if we can find a way to provethat a function

is constantlyincreasingor decreasing,then it isinvertableormonotonic. Frompreviouschapters,you

havelearnedthat if then mustbe increasingand if then mustbe decreasing.

To summarize,a functionhas an inverseif it is one-to-onein its domainor if its derivativeis either
or
Example2:

Giventhe polynomialfunction showthat it is invertable(has an inverse).
Solution:

Takingthe derivative,we find that for all We concludethat is one-to-one
and invertable.Keepin mindthat it may not be easyto find the inverseof (try it!),
but we still knowthat it is indeedinvertable.

Howto find the inverseof a one-to-onefunction:To find the inverseof a one-to-onefunction,simply

solvefor in termsof and then interchange and The resultingformulais the inverse

Example3:

Find the inverseof.
Solution:

Fromthe discussionabove,we can find the inverseby first solvingfor in.

Interchanging ,