Inverse function: Difference between revisions

Revision as of 10:47, 9 September 2023

Revision as of 03:55, 8 November 2023

Line 41:

While the notation {{math|''f''⁻¹(''x'')}} might be misunderstood,<ref name=":2" /> {{math|(''f''(''x''))⁻¹}} certainly denotes the [[multiplicative inverse]] of {{math|''f''(''x'')}} and has nothing to do with the inverse function of {{mvar|f}}.<ref name="Cajori_1929"/> The notation <math>f^{\langle -1\rangle}</math> might be used for the inverse function to avoid ambiguity with the [[multiplicative inverse]].<ref>Helmut Sieber und Leopold Huber: ''Mathematische Begriffe und Formeln für Sekundarstufe I und II der Gymnasien.'' Ernst Klett Verlag.</ref>

In keeping with the general notation, some English authors use expressions like {{math|sin⁻¹(''x'')}} to denote the inverse of the sine function applied to {{mvar|x}} (actually a [[#Partial inverses|partial inverse]]; see below).<ref>{{harvnb|Thomas|1972|loc=pp. 304–309}}</ref><ref name="Cajori_1929"/> Other authors feel that this may be confused with the notation for the multiplicative inverse of {{math|sin (''x'')}}, which can be denoted as {{math|(sin (''x''))⁻¹}}.<ref name="Cajori_1929"/> To avoid any confusion, an [[inverse trigonometric function]] is often indicated by the prefix "[[arc (function prefix)|arc]]" (for Latin {{lang|la|arcus}}).<ref name="Korn_2000"/><ref name="Atlas_2009"/> For instance, the inverse of the sine function is typically called the [[arcsine]] function, written as {{math|[[arcsin]](''x'')}}.<ref name="Korn_2000"/><ref name="Atlas_2009"/> Similarly, the inverse of a [[hyperbolic function]] is indicated by the prefix "[[ar (function prefix)|ar]]" (for Latin {{lang|la|ārea}}).<ref name="Atlas_2009"/> For instance, the inverse of the [[hyperbolic sine]] function is typically written as {{math|[[arsinh]](''x'')}}. <ref name="Atlas_2009"/> The expressions like {{math|sin⁻¹(''x'')}} can still be useful to distinguish the [[Multivalued_functionMultivalued function|multivalued]] inverse from the partial inverse: <math>\sin^{-1}(x) = \{(-1)^n \arcsin(x) + \pi n : n \in \mathbb Z\}</math>. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the {{math|''f''⁻¹}} notation should be avoided.<ref name="Hall_1909"/><ref name="Atlas_2009"/>

== Examples ==

Line 384:

* {{cite book |author-first=Steven R. |author-last=Lay |title=Analysis / With an Introduction to Proof |edition=4 |publisher=[[Pearson (publisher)|Pearson]] / [[Prentice Hall]] |date=2006 |isbn=978-0-13-148101-5 |url=https://books.google.com/books?id=k4k_AQAAIAAJ}}

* {{cite book |author-first1=Douglas |author-last1=Smith |author-first2=Maurice |author-last2=Eggen |author-first3=Richard |author-last3=St. Andre |title=A Transition to Advanced Mathematics |edition=6 |date=2006 |publisher=[[Thompson Brooks/Cole]] |isbn=978-0-534-39900-9}}

* {{cite book |author-first=George Brinton |author-last=Thomas, Jr. |author-link=George Brinton Thomas, Jr. |title=Calculus and Analytic Geometry Part 1: Functions of One Variable and Analytic Geometry |edition=Alternate |date=1972 |publisher=[[Addison-Wesley]] |ref={{SfnRef|Thomas|1972}}}}

* {{cite book |author-first=Robert S. |author-last=Wolf |title=Proof, Logic, and Conjecture / The Mathematician's Toolbox |publisher=[[W. H. Freeman and Co.]] |date=1998 |isbn=978-0-7167-3050-7}}

Line 394:

==External links==

{{sisterlinkssister project links|d=y|b=Algebra/Functions#Inverse function|n=no|c=Category:Inverse function|v=no|voy=no|m=no|mw=no|wikt=inverse function|s=no|species=no|q=no}}

* {{springer|title=Inverse function|id=p/i052360}}

{{Authority control}}

[[Category:Basic concepts in set theory]]