CK12 Calculus - Single Variable

6. TranscendentalFunctions...........................................................................................................

InverseFunctions

Functionssuch as logarithms,exponentialfunctions,and trigonometricfunctionsare examplesoftranscen-
dentalfunctions.If a functionis transcendental,it cannotbe expressedas a polynomialor rationalfunction.
That is, it is not analgebraicfunction.In this chapter, we will beginby developingthe conceptof an inverse
of a functionand how it is linkedto its originalnumerically, algebraically, and graphically. Later, we will take
eachtype of elementarytranscendentalfunction—logarithmic,exponential,and trigonometric—individually
and see the connectionbetweenthemand their respectiveinverses,derivatives,and integrals.

LearningObjectives
A studentwill be able to:

• Understandthe basicpropertiesof the inverseof a functionand how to find it.


• Understandhow a functionand its inverseare representedgraphically.


• Knowthe conditionsof invertabiltyof a function.
  One-to-OneFunctions
  A function,as you knowfrom your previousmathematicsbackground,is a rule that assignsa singlevalue
  in its rangeto eachpointin its domain.In otherwords,for eachoutputnumber, thereis one or moreinput
  numbers.However, a functionneverproducesmorethan a singleoutputfor one input.A functionis said to
  be aone-to-onefunctionif eachoutputis associatedwith only one singleinput.For example,
  assignsthe output for both and and thus it is not aone-to-onefunction.

One-to-OneFunctionThe function is one-to-onein a domain if whenever

Thereis an easymethodto checkif a functionis one-to-one:drawa horizontalline acrossthe graph.If the
line intersectsat only one pointon the graph,then the functionis one-to-one;otherwise,it is not. Noticein
the figurebelowthat the graphof is not one-to-onesincethe horizontalline intersectsthe graph
morethan once.But the function is a one-to-onefunctionbecausethe graphmeetsthe horizontal
line only once.