Inverse function

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In mathematics, the inverse function (or anti-function ) is a function that "inverts" another function: if the function f is applied to input x gives the result y , then implements the inverse function g to y gives the result x , and vice versa. That is, f ( x ) = y if and only if g ( y ) = x .

As a simple example, consider the real function of the real variables given by f ( x ) = 5 x - 7 . Think of this as a step-by-step procedure (ie, take the number x , multiply it by 5, then subtract 7 from the result), to reverse this and get x return of some output value , say y , we have to undo each step in reverse order. In this case it means we have to add 7 to y and then divide the result by 5. In functional notation this inverse function will be given by,

${\ displaystyle g (y) = {\ frac {y 7} {5}}.}$

Dane y = 5 x - 7 kita memiliki f ( x >) = y than g ( y ) = x /span>.

Not all functions have an inverse function. In order for the f function: X -> Y to have the inverse, it must have that property for every y in Y there must be one, and only one x in X so f ( x ) = y . This property ensures that a g function: Y -> X will have the necessary relationships with f .

Video Inverse function

Definition

Biarkan f menjadi fungsi yang domainnya adalah himpunan X , dan yang gambar (rentang) -nya adalah himpunan Y . Kemudian f adalah dapat dibalikkan jika ada fungsi g dengan domain Y dan gambar X , dengan properti:

${\ displaystyle f (x) = y \, \, \ Leftrightarrow \, \, g (y) = x.}  ÂÂ$

If f is reversible, the g function is unique, meaning that there is exactly one function g that satisfies this property (no more, no less). The function g is then called the inverse f , and is usually denoted as f ^-1 .

Expressed otherwise, the function can be reversed if and only if the reverse relation is a function in the range Y , in this case the inverse relation is the inverse function.

Not all functions have an inverse. For a function to have an inverse, any element y ? Y must match no more than one x ? X ; a function f with this property is called one-to-one or injection. If f ^-1 is a function of Y , then every element y Y must match some x ? X . Functions with these properties are called surjections. This property is met by definition if Y is the image (range) of f , but can not apply in the more general context. In order to be reversed, a function must be in the form of injections and surjection. Such functions are referred to as bijections. The opposite of the f : X -> Y is not an ore, that is, a function that is not a surjection, only some functions on Y , which means that for some y ? Y , f ^-1 ( y ) is undefined. If a function of f is invertible, then both and its inverse functions f ^-1 are bijections.

There are other conventions used in function definitions. This can be referred to as a "set-theoretic" or "graph" definition using a sorted pair in which the codomain is never referred. Under this Convention all functions are surjections, and so, being bijection means to be an injection. Authors who use this convention may use the expression that a function may be reversed if and only if it is an injection. Both conventions need not cause confusion as long as it is remembered that in this alternative convention, the codomain of a function is always considered a functional range.

Example: Quadratic and square root functions

The function f : R -> [0 ,?) is given by f ( x ) = < i> x ² not injecting because every possible result y (except 0) corresponds to two different starting points in X - one positive and one negative, and this function can not be reversed. With such a function it is not possible to infer the input of its output. Such functions are called non-injection or, in some applications, loss of information.

If the domain of a function is restricted to a non-negative real, that is, the function is redefined to be f : [0 ,?) -> [0 ,?) with rule the same as before, then the function is pragtip and so on, can be reversed. The inverse function here is called the square root function (positive) .

Inverse and composition

Jika f adalah fungsi yang dapat dibalik dengan domain X dan jangkauan Y , maka

${\ displaystyle f ^ {- 1} \ left (\, f (x) \, \ right) = x} Â$ , untuk setiap ${\ displaystyle x \ di X.} Â Â$

Denounce menggunakan komposisi fungsi, kita dapat menulis ulang pernyataan ini sebagai berikut:

${\ displaystyle f ^ {- 1} \ circ f = \ operatorname {id} _ {X},} Â Â$

where id _X is the identity function on set X ; a function that leaves its argument unchanged. In category theory, this statement is used as the definition of inverse morphism.

Considering the function composition helps understand the f ^-1 notation. Repeatedly arranging functions by itself is called iteration. If f is applied n times, starting with x , then this is written as f ⁿ ( x ) ; so f f ( f (< i> x )) , etc. Since f ^-1 ( f ( x )) = x , compose f ^-1 and f ⁿ generate f ^{n -1} , "undo" the effect of one app f .

Notes on notation

While the f ^-1 ( x ) notes may be misunderstood, ( f ( x )) ^-1 certainly shows the inverse multiplication of f ( x <) and has nothing to do with the inverse function f .

In accordance with the general notation, some English authors use expressions like sin ^-1 ( x ) to indicate the inverse of the applied sine function to x (actually partial opposite; see below) Other authors feel that this may be confusing with the notation for inverse multiplication of sin ( x ) , which can be denoted as (sin ( x )) ^-1 . To avoid confusion, inverted trigonometric functions are often indicated by the "arc" prefix (for Latin arcus ). For example, the inverse of the sine function is usually called the arcsine function, written as arcsin ( x ) . Similarly, the inverse of the hyperbolic function is indicated by the prefix "ar" (for Latin area ). For example, the inverse of the hyperbolic sine function is usually written as arsinh ( x ) . Other inverted special functions are sometimes preceded by the "inv" prefix if the notation ambiguity f ^-1 should be avoided.

Maps Inverse function

Properties

Since a function is a special type of binary relation, many properties of the inverse function are related to the properties of the inverse relationship.

Uniqueness

If the inverse function exists for a given function f , then it is unique. This follows because the inverse function must be a fully inverse relation defined by f .

Symmetry

There is symmetry between function and inverse. Specifically, if f is a reversible function with the domain X and the range Y , then its invers f ^-1 has domain Y and range X , and invers f ^-1 is the original function f . In a symbol, for the f : X -> Y and f ¹ : Y -> X ,

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âf^{Â\; Â\; Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ,Â-Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ\; 1\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â?Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂfÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          en                        X                                {\ displaystyle f ^ {- 1} \ circ f = \ operatorname {id} _ {X}}  Â} and $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂfÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â?Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âf^{Â\; Â\; Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ,Â-Â\; Â\; ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ\; 1\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          en                        Y                          .               {\ displaystyle f \ circ f ^ {- 1} = \ operatorname {id} _ {Y}.}  Â}

Pernyataan ini merupakan konsekuensi dari implikasi bahwa untuk f agar dapat dibalik, itu harus bersifat bijective. Sifat involuntario dari invers dapat diezpresikan secara singkat oleh

${\ displaystyle \ left (f ^ {- 1} \ right) ^ {- 1} = f.} Â Â$

Inversi dari komposisi fungsi diberikan oleh

${\ displaystyle (g \ circ f) ^ {- 1} = f ^ {- 1} \ circ g ^ {- 1}.} Â Â$

Note that the order of g and f has been reversed; to cancel f g , we must first undo g and then cancel f .

Misalnya, beri f ( x ) = 3 x dan beri g ( x ) = x 5 . Lalu komposisinya g ? f adalah fungsi yang pertama kali mengalikan dengan tiga dan kemudian menambahkan lima,

${\ displaystyle (g \ circ f) (x) = 3x 5.}  ÂÂ$

Untuk membalikkan proses ini, pertama-tama kita harus mengurangi lima, dan kemudian membagi dengan tiga,

${\ displaystyle (g \ circ f) ^ {- 1} (x) = {\ tfrac {1} {3}} (x-5). } Â Â$

This is the composition of ( f ^-1 ? g ^-1 i>) .

Conversions yourself

Jika X adalah satu set, maka fungsi identify all X adalah kebalikannya send:

${\ displaystyle {\ operatorname {id} _ {X}} ^ {- 1} = \ operatorname {id} _ {X}.} Â Â$

More generally, the function f Ã,: X -> X is equal to its own inverse if and only if the composition < span> f ? f id _X . Such a function is called involution.

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Inverted in calculus

Kalkulus variabel tunggal terutama berkaitan dengan fungsi-fungsi yang memetakan bilangan real that bilangan real. Fungi seperti itu sering didefinisikan melalui rumus, seperti:

${\ displaystyle f (x) = (2x 8) ^ 3}. Â Â$

The function of surjective f from the real number to the real number has an inversion provided one-to-one, eg along the chart y = f (< i> x ) has, for any y value that may be just one corresponding x value, and thus passes the horizontal line test.

The following table shows some of its standard functions and inversions:

The formula for inverse

Satu pendekatan untuk menemukan rumus untuk f ^-1 , jika ada, adalah untuk menyelesaikan persamaan and = f ( x ) untuk x . Misalnya, jika f adalah fungsinya

${\ displaystyle f (x) = (2x 8) ^ 3}} Â Â$

maka kita harus menyelesaikan persamaan y = (2 x 8) ³ untuk x :

${\ displaystyle {\ begin {aligned} y & amp; = (2x 8) ^ {3} \\ {\ sqrt [{3}] {y}} & amp; = 2x 8 \\ {\ sqrt [{3}] {y}} - 8 & amp; = 2x \\ {\ dfrac {{\ sqrt [{3}] {y}} - 8} {2}} & amp; = x. \ end {aligned}}}  ÂÂ$

Denuncia demikian fungsi inverse f ^-1 diberikan oleh rumus

${\ displaystyle f ^ {- 1} (y) = {\ frac {{\ sqrt [{3}] {y}} - 8} {2 }}.} Â Â$

Terkadang Cuban fungi tidak dapat diexpressive denounced rumus dense sejumlah istilah terbatas. Misalnya, jika f adalah fungsinya

${\ displaystyle f (x) = x- \ sin x,} Â Â$

maka f adalah sebuah bijion, dan karena itu memiliki fungsi inverse f ^-1 . Rumus untuk invers ini memiliki jumlah istilah tak terbatas:

${\ displaystyle f ^ {- 1} (y) = \ jumlah _ {n = 1} ^ {\ infty} {\ frac {y ^ {n/3}} {n!}} \ lim _ {\ theta \ to 0} \ left ({\ frac {\ mathrm {d} ^ {\, n-1}} {\ mathrm {d} \ theta ^ {\, n- 1}}} \ kiri ({\ frac {\ theta} {\ sqrt [{3}] {\ theta - \ sin (\ theta)}}} \ right) ^ {n} \ right).}  ÂÂ$

Grafik dari inverse

Jika f dapat dibalik, maka grafik fungsi

${\ displaystyle y = f ^ {- 1} (x)} Â Â$

same denial graphic persuasion

${\ displaystyle x = f (y).} Â Â$

This is identical to the y = f ( x ) equation which defines the graph f , except that the role of x and y has been reversed. Thus the f ^-1 graph can be obtained from the f graph by switching the position x and y axes. This is equivalent to reflecting the graph across the line y = x .

Reversals and descending

Fungi kontiny f dapat dibalik pada jangkauannya (gambar) jika dan hanya jika itu benar-benar meningkat atau menurun secara drastis (tanpa maksimum lokal atau minimum). Misalnya, fungi

${\ displaystyle f (x) = x ^ {3} x} Â Â$

can be reversed, since the f? ( x ) = 3 x ² 1 is always positive.

Jika fungsi f dapat terdiferensiasi pada interval I dan f? ( x )? 0 untuk masing-masing x ? Saya , maka inverse f ^-1 akan terdiferensiasi pada f ( I ) . Jika y = f ( x ) , turunan dari invers diberikan oleh teorema fungsi invers,

${\ displaystyle \ left (f ^ {- 1} \ right) ^ {\ prime} (y) = {\ frac {1} {f '\ left (x \ kanan)}}.}  ÂÂ$

Menggunakan notasi Leibniz, rumus di atas dapat ditulis sebagai

${\ displaystyle {\ frac {dx} {dy}} = {\ frac {1} {dy/dx}}.} Â Â$

These results follow the chain rule (see article on inverse function and differentiation).

Bahkan jika fungsi f tidak satu-ke-satu, dimungkinkan untuk menentukan kebalikan parsial dari f dengan membatasi domain. Misalnya, fungi

${\ displaystyle f (x) = x ^ 2} Â Â$

bukan satu-ke-satu, karena x ² = (- x ) ² . Namun, fungsi menjadi satu-ke-satu jika kita membatasi that domain x > = 0 , dalam hal ini

${\ displaystyle f ^ {- 1} (y) = {\ sqrt {y}}.} Â Â$

(Jika kita malah membatasi that domain x <= 0 , maka invers adalah negatif dari akar kuadrat and .) Atau, tidak perlu membatasi domain jika kita puas dengan invers yang menjadi fungsi multinilai:

${\ displaystyle f ^ {- 1} (y) = \ pm {\ sqrt {y}}.} Â Â$

Sometimes the inverse multinilai is called full reverse of f , and its parts (like ? x and - ? x ) are called branch . The most important branch of a multinilai function (eg a positive square root) is called main branch , and its value in y is called the primary value of < i> f ^-1 ( y ) .

For a continuous function on a real line, one branch is required between each local extreme pair. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see adjacent image).

Pertimbangan ini has been killed by a single-level mendefinisan inverse fungi trigonometry. Misalnya, fungsi sinus tidak satu-ke-satu, karena

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