Inmathematics, theinverse function of afunctionf (also called theinverse off) is afunction that undoes the operation off. The inverse off existsif and only iff isbijective, and if it exists, is denoted by
For a function, its inverse admits an explicit description: it sends each element to the unique element such thatf(x) =y.
As an example, consider thereal-valued function of a real variable given byf(x) = 5x − 7. One can think off as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse off is the function defined by
Letf be a function whosedomain is thesetX, and whosecodomain is the setY. Thenf isinvertible if there exists a functiong fromY toX such that for all and for all.[1]
Iff is invertible, then there is exactly one functiong satisfying this property. The functiong is called the inverse off, and is usually denoted asf −1, a notation introduced byJohn Frederick William Herschel in 1813.[2][3][4][5][6][nb 1]
The functionf is invertible if and only if it is bijective. This is because the condition for all implies thatf isinjective, and the condition for all implies thatf issurjective.
The inverse functionf −1 tof can be explicitly described as the function
Recall that iff is an invertible function with domainX and codomainY, then
, for every and for every.
Using thecomposition of functions, this statement can be rewritten to the following equations between functions:
and
whereidX is theidentity function on the setX; that is, the function that leaves its argument unchanged. Incategory theory, this statement is used as the definition of an inversemorphism.
Considering function composition helps to understand the notationf −1. Repeatedly composing a functionf:X→X with itself is callediteration. Iff is appliedn times, starting with the valuex, then this is written asfn(x); sof 2(x) =f (f (x)), etc. Sincef −1(f (x)) =x, composingf −1 andfn yieldsfn−1, "undoing" the effect of one application off.
While the notationf −1(x) might be misunderstood,[1](f(x))−1 certainly denotes themultiplicative inverse off(x) and has nothing to do with the inverse function off.[6] The notation might be used for the inverse function to avoid ambiguity with themultiplicative inverse.[7]
In keeping with the general notation, some English authors use expressions likesin−1(x) to denote the inverse of the sine function applied tox (actually apartial inverse; see below).[8][6] Other authors feel that this may be confused with the notation for the multiplicative inverse ofsin (x), which can be denoted as(sin (x))−1.[6] To avoid any confusion, aninverse trigonometric function is often indicated by the prefix "arc" (for Latinarcus).[9][10] For instance, the inverse of the sine function is typically called thearcsine function, written asarcsin(x).[9][10] Similarly, the inverse of ahyperbolic function is indicated by the prefix "ar" (for Latinārea).[10] For instance, the inverse of thehyperbolic sine function is typically written asarsinh(x).[10] The expressions likesin−1(x) can still be useful to distinguish themultivalued inverse from the partial inverse:. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of thef −1 notation should be avoided.[11][10]
The functionf:R → [0,∞) given byf(x) =x2 is not injective because for all. Therefore,f is not invertible.
If the domain of the function is restricted to the nonnegative reals, that is, we take the function with the samerule as before, then the function is bijective and so, invertible.[12] The inverse function here is called the(positive) square root function and is denoted by.
Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse of an invertible function has an explicit description as
.
This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, iff is the function
then to determine for a real numbery, one must find the unique real numberx such that(2x + 8)3 =y. This equation can be solved:
Thus the inverse functionf −1 is given by the formula
Sometimes, the inverse of a function cannot be expressed by aclosed-form formula. For example, iff is the function
thenf is a bijection, and therefore possesses an inverse functionf −1. Theformula for this inverse has an expression as an infinite sum:
If an inverse function exists for a given functionf, then it is unique.[13] This follows since the inverse function must be the converse relation, which is completely determined byf.
There is a symmetry between a function and its inverse. Specifically, iff is an invertible function with domainX and codomainY, then its inversef −1 has domainY and imageX, and the inverse off −1 is the original functionf. In symbols, for functionsf:X →Y andf−1:Y →X,[13]
and
This statement is a consequence of the implication that forf to be invertible it must be bijective. Theinvolutory nature of the inverse can be concisely expressed by[14]
The inverse ofg ∘ f isf −1 ∘ g −1.
The inverse of a composition of functions is given by[15]
Notice that the order ofg andf have been reversed; to undof followed byg, we must first undog, and then undof.
For example, letf(x) = 3x and letg(x) =x + 5. Then the compositiong ∘ f is the function that first multiplies by three and then adds five,
To reverse this process, we must first subtract five, and then divide by three,
More generally, a functionf :X →X is equal to its own inverse, if and only if the compositionf ∘ f is equal toidX. Such a function is called aninvolution.
The graphs ofy =f(x) andy =f −1(x). The dotted line isy =x.
Iff is invertible, then the graph of the function
is the same as the graph of the equation
This is identical to the equationy =f(x) that defines the graph off, except that the roles ofx andy have been reversed. Thus the graph off −1 can be obtained from the graph off by switching the positions of thex andy axes. This is equivalent toreflecting the graph across the liney =x.[16][1]
is invertible, since thederivativef′(x) = 3x2 + 1 is always positive.
If the functionf isdifferentiable on an intervalI andf′(x) ≠ 0 for eachx ∈I, then the inversef −1 is differentiable onf(I).[17] Ify =f(x), the derivative of the inverse is given by the inverse function theorem,
The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiablemultivariable functionf:Rn →Rn is invertible in a neighborhood of a pointp as long as theJacobian matrix off atp isinvertible. In this case, the Jacobian off −1 atf(p) is thematrix inverse of the Jacobian off atp.
Letf be the function that converts a temperature in degreesCelsius to a temperature in degreesFahrenheit, then its inverse function converts degrees Fahrenheit to degrees Celsius,[18] since
Supposef assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has children born in the same year (for instance, twins or triplets, etc.) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example,
LetR be the function that leads to anx percentage rise of some quantity, andF be the function producing anx percentage fall. Applied to $100 withx = 10%, we find that applying the first function followed by the second does not restore the original value of $100, demonstrating the fact that, despite appearances, these two functions are not inverses of each other.
The formula to calculate the pH of a solution ispH = −log10[H+]. In many cases we need to find the concentration of acid from a pH measurement. The inverse function[H+] = 10−pH is used.
Alternatively, there is no need to restrict the domain if we are content with the inverse being amultivalued function:
Sometimes, this multivalued inverse is called thefull inverse off, and the portions (such as√x and −√x) are calledbranches. The most important branch of a multivalued function (e.g. the positive square root) is called theprincipal branch, and its value aty is called theprincipal value off −1(y).
For a continuous function on the real line, one branch is required between each pair oflocal extrema. For example, the inverse of acubic function with a local maximum and a local minimum has three branches (see the adjacent picture).
Thearcsine is a partial inverse of thesine function.
The above considerations are particularly important for defining the inverses oftrigonometric functions. For example, thesine function is not one-to-one, since
for every realx (and more generallysin(x + 2πn) = sin(x) for everyintegern). However, the sine is one-to-one on the interval[−π/2, π/2], and the corresponding partial inverse is called thearcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −π/2 andπ/2. The following table describes the principal branch of each inverse trigonometric function:[19]
Function composition on the left and on the right need not coincide. In general, the conditions
"There existsg such thatg(f(x))=x" and
"There existsg such thatf(g(x))=x"
imply different properties off. For example, letf:R →[0, ∞) denote the squaring map, such thatf(x) =x2 for allx inR, and letg:[0, ∞) →R denote the square root map, such thatg(x) =√x for allx ≥ 0. Thenf(g(x)) =x for allx in[0, ∞); that is,g is a right inverse tof. However,g is not a left inverse tof, since, e.g.,g(f(−1)) = 1 ≠ −1.
Iff:X →Y, aleft inverse forf (orretraction off ) is a functiong:Y →X such that composingf withg from the left gives the identity function[20] That is, the functiong satisfies the rule
Iff(x)=y, theng(y)=x.
The functiong must equal the inverse off on the image off, but may take any values for elements ofY not in the image.
A functionf with nonempty domain is injective if and only if it has a left inverse.[21] An elementary proof runs as follows:
Ifg is the left inverse off, andf(x) =f(y), theng(f(x)) =g(f(y)) =x =y.
If nonemptyf:X →Y is injective, construct a left inverseg:Y →X as follows: for ally ∈Y, ify is in the image off, then there existsx ∈X such thatf(x) =y. Letg(y) =x; this definition is unique becausef is injective. Otherwise, letg(y) be an arbitrary element ofX.
For allx ∈X,f(x) is in the image off. By construction,g(f(x)) =x, the condition for a left inverse.
In classical mathematics, every injective functionf with a nonempty domain necessarily has a left inverse; however, this may fail inconstructive mathematics. For instance, a left inverse of theinclusion{0,1} →R of the two-element set in the reals violatesindecomposability by giving aretraction of the real line to the set{0,1}.[22]
Example ofright inverse with non-injective, surjective function
Aright inverse forf (orsection off ) is a functionh:Y →X such that
That is, the functionh satisfies the rule
If, then
Thus,h(y) may be any of the elements ofX that map toy underf.
A functionf has a right inverse if and only if it issurjective (though constructing such an inverse in general requires theaxiom of choice).
Ifh is the right inverse off, thenf is surjective. For all, there is such that.
Iff is surjective,f has a right inverseh, which can be constructed as follows: for all, there is at least one such that (becausef is surjective), so we choose one to be the value ofh(y).[23]
An inverse that is both a left and right inverse (atwo-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be calledthe inverse.
If is a left inverse and a right inverse of, for all,.
A function has a two-sided inverse if and only if it is bijective.
A bijective functionf is injective, so it has a left inverse (iff is the empty function, is its own left inverse).f is surjective, so it has a right inverse. By the above, the left and right inverse are the same.
Iff has a two-sided inverseg, theng is a left inverse and right inverse off, sof is injective and surjective.
Iff:X →Y is any function (not necessarily invertible), thepreimage (orinverse image) of an elementy ∈Y is defined to be the set of all elements ofX that map toy:
The preimage ofy can be thought of as theimage ofy under the (multivalued) full inverse of the functionf.
The notion can be generalized to subsets of the range. Specifically, ifS is anysubset ofY, the preimage ofS, denoted by, is the set of all elements ofX that map toS:
For example, take the functionf:R →R;x ↦x2. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.
.
The original notion and its generalization are related by the identity The preimage of a single elementy ∈Y – asingleton set{y} – is sometimes called thefiber ofy. WhenY is the set of real numbers, it is common to refer tof −1({y}) as alevel set.
^Peirce, Benjamin (1852).Curves, Functions and Forces. Vol. I (new ed.). Boston, USA. p. 203.{{cite book}}: CS1 maint: location missing publisher (link)
^Peano, Giuseppe (1903).Formulaire mathématique (in French). Vol. IV. p. 229.
^abcdCajori, Florian (1952) [March 1929]. "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions".A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA:Open court publishing company. pp. 108,176–179, 336, 346.ISBN978-1-60206-714-1. Retrieved2016-01-18.[...] §473.Iterated logarithms [...] We note here the symbolism used byPringsheim andMolk in their jointEncyclopédie article: "2logba = logb (logba), ...,k+1logba = logb (klogba)." [...] §533.John Herschel's notation for inverse functions, sin−1x, tan−1x, etc., was published by him in thePhilosophical Transactions of London, for the year 1813. He says (p. 10): "This notation cos.−1e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.mA for (cos.A)m, but he justifies his own notation by pointing out that sinced2x, Δ3x, Σ2x meanddx, ΔΔΔx, ΣΣx, we ought to write sin.2x for sin. sin.x, log.3x for log. log. log.x. Just as we writed−n V=∫n V, we may write similarly sin.−1x=arc (sin.=x), log.−1x.=cx. Some years later Herschel explained that in 1813 he usedfn(x),f−n(x), sin.−1x, etc., "as he then supposed for the first time. The work of a German Analyst,Burmann, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."[a] [...] §535.Persistence of rival notations for inverse function.— [...] The use of Herschel's notation underwent a slight change inBenjamin Peirce's books, to remove the chief objection to them; Peirce wrote: "cos[−1]x," "log[−1]x."[b] [...] §537.Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sinx, namely, (sinx)2, sinx2, sin2x. The prevailing notation at present is sin2x, though the first is least likely to be misinterpreted. In the case of sin2x two interpretations suggest themselves; first, sinx · sinx; second,[c] sin (sinx). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2x, where logx · logx and log (logx) are of frequent occurrence in analysis. [...] The notation sinnx for (sinx)n has been widely used and is now the prevailing one. [...] (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
^Helmut Sieber und Leopold Huber:Mathematische Begriffe und Formeln für Sekundarstufe I und II der Gymnasien. Ernst Klett Verlag.
^Hall, Arthur Graham; Frink, Fred Goodrich (1909)."Article 14: Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA.Plane Trigonometry. New York:Henry Holt & Company. pp. 15–16. Retrieved2017-08-12.α = arcsin m This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sinm is perhaps better still on account of its general applicability. [...] A similar symbolic relation holds for the othertrigonometric functions. It is frequently read 'arc-sinem' or 'anti-sinem', since two mutually inverse functions are said each to be the anti-function of the other.
