|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{inline citations|date=May 2013}}
| | Gold jewelry is a look that"s never gone out of fashion and is as popular as ever. Browse this web site [https://twitter.com/goldbackediras get gold backed ira] to learn how to consider it. Purer silver jewelry doesn"t cause staining or tarnishing like poor alloy combinations. Usually when someone includes a problem with discoloration on their skin it"s for their human body and not the silver. Visiting [https://twitter.com/goldpricesnow gold price investigation] likely provides lessons you can tell your family friend. <br><br> |
| {{Technical|date=May 2011}}
| |
| [[Image:Inverse Function.png|thumb|right|A function {{mvar|f}} and its inverse {{math|''f''<sup> −1</sup>}}. Because {{mvar|f}} maps {{mvar|a}} to 3, the inverse {{math|''f''<sup> −1</sup>}} maps 3 back to {{mvar|a}}.]]
| |
| {{Functions}}
| |
| In [[mathematics]], an '''inverse function''' is a [[function (mathematics)|function]] that reverses another function: if the function {{mvar|f}} applied to an input {{mvar|x}} gives a result of {{mvar|y}}, then applying the inverse function {{mvar|g}} to {{mvar|y}} gives the result {{mvar|x}}, and vice versa. i.e. {{math|1=''f''(''x'') = ''y''}}, and {{math|1=''g''(''y'') = ''x''}}. More directly, {{math|1=''g''(''f''(''x'')) = ''x''}}, meaning {{mvar|g}} [[Function composition|composed]] with {{mvar|f}} form an [[identity function|identity]].
| |
|
| |
|
| A function {{mvar|f}} that has an inverse is defined as '''invertible'''; the inverse function is then uniquely determined by {{mvar|f}} and is denoted by {{math|''f''<sup> −1</sup>}}, read ''f inverse''. [[Superscript]]ed "{{num|−1}}" does not refer to [[number|numerical]] [[exponentiation]]: see [[composition monoid]] for explanation of this notation.
| | Gold is extremely flexible and as such can be changed to just about any shape including little hair like strands and thin sheets. <br><br>One of the main things most people search for when buying silver jewelry is the purity. The jewelry industry features a common system for distinguishing this element. <br><br>Odds are the ring on your hand is marked 18K, 14K, or 10K, with the K standing for karat, the system used to explain the proportion of pure gold an item contains. <br><br>The higher the karat number, the higher the proportion of gold in your gold jewelry. <br><br>24K gold is pure gold. <br><br>18K gold contains 18 parts gold and 6 parts of one or maybe more additional materials, rendering it 75% gold. <br><br>14K gold contains 14 parts gold and 10 parts of 1 or even more additional metals, rendering it 58.3% gold. <br><br>12K gold contains 12 parts gold and 12 parts of just one or even more additional materials, making it 50% gold. <br><br>10K gold includes 10 parts gold and 14 parts of 1 or maybe more additional materials, which makes it 41.7% gold. 10K gold could be the minimum karat which can be called "gold" in the Usa. <br><br>European gold jewelry is marked with numbers that show their proportion of gold, such as: <br><br>18K gold is marked 750 to point 75% gold <br><br>14K silver is marked 585 for 58.5% <br><br>12K silver is noted 417 for 41.7% <br><br>The karat marking in your gold jewelry must certanly be with a characteristic or brand that identifies its producer. The item"s country of origin could also be involved. <br><br>You will find samples of pure gold jewellery, but pure gold is soft and isn"t practical for everyday wear. Other materials are combined with it to create it more durable (and to lower its price). <br><br>[http://Www.Alexa.com/search?q=Strong+silver&r=topsites_index&p=bigtop Strong silver] is tough, so it is an improved choice for jewelry you"ll wear regularly. If you have allergies to nickel or other materials, choose things that have high gold content, such as for example 18K or 22K gold jewelry..Regal Assets<br>2600 W Olive Ave, Burbank, CA 91505<br><br>If you adored this article along with you would want to acquire guidance relating to [http://www.blogigo.com/lavishtechnique54 health promotion] i implore you to visit our own web-site. |
| | |
| ==Definitions==
| |
| [[Image:Inverse Functions Domain and Range.png|thumb|right|240px|If {{mvar|f}} maps {{mvar|X}} to {{mvar|Y}}, then {{math|''f''<sup> −1</sup>}} maps {{mvar|Y}} back to {{mvar|X}}.]]
| |
| {{See also|Inverse element}}
| |
| | |
| Instead of considering the inverses for individual inputs and outputs, one can think of the function as sending the whole set of inputs, the [[domain of a function|domain]], to a set of outputs, the [[Range (mathematics)|range]]. Let {{mvar|f}} be a function whose domain is the [[Set (mathematics)|set]] {{mvar|X}}, and whose range is the set {{mvar|Y}}. Then {{mvar|f}} is invertible if there exists a function {{mvar|g}} with domain {{mvar|Y}} and range {{mvar|X}}, with the property:
| |
| :<math>f(x) = y\,\,\Leftrightarrow\,\,g(y) = x.</math>
| |
| | |
| If {{mvar|f}} is invertible, the function {{mvar|g}} is [[uniqueness|unique]]; in other words, there is exactly one function {{mvar|g}} satisfying this property (no more, no less). That function {{mvar|g}} is then called ''the'' inverse of {{mvar|f}}, and usually denoted as {{math|''f''<sup> −1</sup>}}.
| |
| | |
| Stated otherwise, a function is invertible if and only if its [[inverse relation]] is a function on the range {{mvar|Y}}, in which case the inverse relation is the inverse function.{{citation needed|date=August 2013}}
| |
| | |
| Not all functions have an inverse. For this rule to be applicable, each element {{math|''y'' ∈ ''Y''}} must correspond to no more than one {{math|''x'' ∈ ''X''}}; a function {{mvar|f}} with this property is called one-to-one, or information-preserving, or an [[injective function|injection]]. If {{mvar|f}} and {{math|''f''<sup> −1</sup>}} are [[total function]]s on {{mvar|X}} and {{mvar|Y}} respectively, then both are [[bijection]]s. The inverse of an injection that is not a bijection is a [[partial function]], that means for some {{math|''y'' ∈ ''Y''}} it is undefined.
| |
| | |
| ===Example: inverse operations that lead to inverse functions===
| |
| Inverse operations are the opposite of direct variation functions. Direct variation functions are based on multiplication; {{math|1=''y'' = ''kx''}}. The opposite operation of multiplication is division and an inverse variation function is {{math|1=''x'' = ''y''/''k''}}.
| |
| | |
| ===Example: percentages===
| |
| Despite their familiarity, percentage changes do not have a straightforward inverse. That is, a X per cent fall is not the inverse of an X per cent rise.
| |
| | |
| ===Example: squaring and square root functions===
| |
| The function {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} may or may not be invertible, depending on what kinds of numbers are being considered (the "domain").
| |
| | |
| If the domain is the [[real number]]s, then each possible result ''y'' corresponds to two different starting points in {{mvar|X}}: one positive and one negative ({{math|[[sign (mathematics)|±]]''x''}}), and so this function is not invertible: as it is impossible to deduce from its output the sign of its input. Such a function is called non-[[Injective function|injective]] or information-losing. Neither the [[square root]] nor the [[principal square root]] function is the inverse of {{math|''x''<sup>2</sup>}} because the first is not [[Single-valued function|single-valued]], and the second returns [[additive inverse|{{math|−''x''}}]] when {{mvar|x}} is negative.
| |
| | |
| If only positive numbers (and zero) are being considered, then the function is injective and invertible.
| |
| | |
| ===Inverses in higher mathematics===
| |
| The definition given above is commonly adopted in [[set theory]] and [[calculus]]. In higher mathematics, the notation | |
| :<math>f\colon X \to Y </math>
| |
| means "{{mvar|f}} is a function mapping elements of a set {{mvar|X}} to elements of a set {{mvar|Y }}". The source, {{mvar|X}}, is called the domain of {{mvar|f}}, and the target, {{mvar|Y}}, is called the [[codomain]]. The codomain contains the range of {{mvar|f}} as a [[subset]], and is considered{{by whom|date=July 2013}} part of the definition of {{mvar|f}}.
| |
| | |
| When using codomains, the inverse of a function {{math| ''f'': ''X'' → ''Y''}} is required to have domain {{mvar|Y}} and codomain {{mvar|X}}. For the inverse to be defined on all of {{mvar|Y}}, every element of {{mvar|Y}} must lie in the range of the function {{mvar|f}}. A function with this property is called ''onto'' or a ''[[Surjective function|surjection]]''. Thus, a function with a codomain is invertible [[if and only if]] it is both injective (one-to-one) and surjective (onto). Such a function is called a one-to-one correspondence or a [[bijection]], and has the property that every element {{math| ''y'' ∈ ''Y''}} corresponds to exactly one element {{math| ''x'' ∈ ''X''}}.
| |
| | |
| ===Inverses and composition===
| |
| If {{mvar|f}} is an invertible function with domain {{mvar|X}} and range {{mvar|Y}}, then
| |
| | |
| :<math> f^{-1}\left( \, f(x) \, \right) = x</math>, for every <math>\displaystyle x \in X. </math>
| |
| | |
| This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if {{mvar|Y}} coincides with the codomain of {{mvar|f}}. Using the [[composition of functions]] we can rewrite this statement as follows:
| |
| | |
| :<math> f^{-1} \circ f = \mathrm{id}_X, </math>
| |
| | |
| where {{math|id<sub>''X''</sub>}} is the [[identity function]] on the set {{mvar|X}}; that is, the function that leaves its argument unchanged. In [[category theory]], this statement is used as the definition of an inverse [[morphism]].
| |
| | |
| If we think of composition as a kind of multiplication{{vague|date=July 2013|reason=Why not to say something about rings, monoids, or groups?}} of functions, this identity says that the inverse of a function is analogous to a [[multiplicative inverse]]. This explains the origin of the notation {{math|''f''<sup> −1</sup>}}. A similar notation is used for [[iterated function]]s.
| |
| | |
| === Note on notation ===
| |
| | |
| The superscript notation for inverses can sometimes be confused with other uses of superscripts, especially when dealing with [[Trigonometric functions|trigonometric]] and [[hyperbolic function|hyperbolic]] functions. To avoid this confusion, the notations {{math|''f''<sup> [−1]</sup>}} or with the "{{math|<sup>−1</sup>}}" above the {{mvar|f}} are sometimes used.{{Citation needed|date=January 2009}}
| |
| | |
| Whereas the notation {{math|''f''<sup> −1</sup>(''x'')}} might be ambiguous, {{math|''f''(''x'')<sup>−1</sup>}} certainly denotes the [[multiplicative inverse]] of {{math|''f''(''x'')}} and has nothing to with inversion of {{mvar|f}}.
| |
| | |
| The expression {{math|sin<sup>−1</sup> ''x''}} usually{{fact|date=July 2013}} does not represent the multiplicative inverse to {{math|sin ''x''}}, but the inverse of the sine function applied to {{mvar|x}} (actually a [[#Partial inverses|partial inverse]]; see below). To avoid confusion, an [[inverse trigonometric function]] is often indicated by the prefix "arc". For instance, the inverse of the sine function is typically called the [[arcsine]] function, written as arcsin, which is, like sin, conventionally denoted in [[roman type]] and not in [[italics]] (note that software libraries of mathematical functions often use the name <tt>asin</tt>):
| |
| | |
| :<math>\sin^{-1} x = \arcsin x.</math>
| |
| | |
| The function {{math|(sin ''x'')<sup> −1</sup>}} is the multiplicative inverse to the sine, and is called the [[cosecant]]. It is usually{{fact|date=July 2013}} denoted {{math|csc ''x''}}:
| |
| :<math> \csc x = (\sin x)^{-1} = \frac{1}{\sin x}.</math>
| |
| | |
| Hyperbolic functions behave similarly, using the prefix "ar" for [[inverse hyperbolic function|their inverse functions]], as in arsinh for the inverse function of sinh, and {{math|csch ''x''}} for the multiplicative inverse of {{math|sinh ''x''}}.
| |
| | |
| ==Properties==
| |
| ===Uniqueness===
| |
| If an inverse function exists for a given function {{mvar|f}}, it is unique: it must be the [[inverse relation]].
| |
| | |
| ===Symmetry===
| |
| There is a symmetry between a function and its inverse. Specifically, if {{mvar|f}} is an invertible function with domain {{mvar|X}} and range {{mvar|Y}}, then its inverse {{math|''f''<sup> −1</sup>}} has domain {{mvar|Y}} and range {{mvar|X}}, and the inverse of {{math|''f''<sup> −1</sup>}} is the original function {{mvar|f}}. In symbols, for {{mvar|f}} a function with domain {{mvar|X}} and range {{mvar|Y}}, and {{mvar|g}} a function with domain {{mvar|Y}} and range {{mvar|X}}:
| |
| | |
| :<math>\begin{align}
| |
| &\text{If } &g \circ f = \mathrm{id}_X\text{,} \\
| |
| &\text{then } &f \circ g = \mathrm{id}_Y\text{.}
| |
| \end{align}</math>
| |
| | |
| This follows from the connection between function inverse and relation inverse, because [[inverse relation|inversion of relations]] is an [[involution (mathematics)|involution]].
| |
| | |
| This statement is an obvious consequence of the deduction that for {{mvar|f}} to be invertible it must be injective (first definition of the inverse) or bijective (second definition). The property of [[involution (mathematics)|involutive symmetry]] can be concisely expressed by the following formula:
| |
| | |
| :<math>\left(f^{-1}\right)^{-1} = f .</math>
| |
| | |
| [[Image:Composition of Inverses.png|thumb|right|240px|The inverse of {{math| ''g'' ∘ ''f'' }} is {{math| ''f''<sup> −1</sup> ∘ ''g''<sup> −1</sup>}}.]]
| |
| The inverse of a composition of functions is given by the formula
| |
| :<math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>
| |
| Notice that the order of {{mvar|g}} and {{mvar|f}} have been reversed; to undo {{mvar|f}} followed by {{mvar|g}}, we must first undo {{mvar|g}} and then undo {{mvar|f}}.
| |
| | |
| For example, let {{math|1= ''f''(''x'') = 3''x''}} and let {{math|1= ''g''(''x'') = ''x'' + 5}}. Then the composition {{math| ''g'' ∘ ''f''}} is the function that first multiplies by three and then adds five:
| |
| :<math>(g \circ f)(x) = 3x + 5</math> | |
| To reverse this process, we must first subtract five, and then divide by three:
| |
| :<math>(g \circ f)^{-1}(y) = \tfrac13(y - 5)</math>
| |
| This is the composition
| |
| {{math| (''f''<sup> −1</sup> ∘ ''g''<sup> −1</sup>)(''y'')}}.
| |
| | |
| ===Self-inverses===
| |
| If {{mvar|X}} is a set, then the [[identity function]] on {{mvar|X}} is its own inverse:
| |
| | |
| :<math>\mathrm{id}_X^{-1} = \mathrm{id}_X</math>
| |
| | |
| More generally, a function {{math| ''f'' : ''X'' → ''X''}} is equal to its own inverse if and only if the composition {{math| ''f'' ∘ ''f''}} is equal to {{math|id<sub>''X''</sub>}}. Such a function is called an [[involution (mathematics)|involution]].
| |
| | |
| ==Inverses in calculus==
| |
| Single-variable [[calculus]] is primarily concerned with functions that map [[real number]]s to real numbers. Such functions are often defined through [[formula]]s, such as:
| |
| :<math>f(x) = (2x + 8)^3 .</math>
| |
| A function {{mvar|f}} from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as long as the graph of {{math|1=''y'' = ''f''(''x'')}} has, for each possible {{mvar|y}} value only one corresponding {{mvar|x}} value, and thus passes the [[horizontal line test]].
| |
| | |
| The following table shows several standard functions and their inverses:
| |
| :{| class="wikitable" align="center"
| |
| |-
| |
| !align="center"| Function {{math|''f''(''x'')}}
| |
| !align="center"| Inverse {{math|''f''<sup> −1</sup>(''y'')}}
| |
| !align="center"| Notes
| |
| |-
| |
| |align="center"| {{math|''x'' [[addition|+]] ''a''}}
| |
| |align="center"| {{math|''y'' [[subtraction|−]] ''a''}}
| |
| |
| |
| |-
| |
| |align="center"| {{math|''a'' − ''x''}}
| |
| |align="center"| {{math|''a'' − ''y''}}
| |
| |
| |
| |-
| |
| |align="center"| [[multiplication|{{math|''mx''}}]]
| |
| |align="center"| [[division (mathematics)|{{sfrac|{{mvar|y}}|{{mvar|m}}}}]]
| |
| | {{math|''m'' ≠ 0}}
| |
| |-
| |
| |align="center"| {{sfrac|1|{{mvar|x}}}}
| |
| |align="center"| {{sfrac|1|{{mvar|y}}}}
| |
| | {{math|''x'', ''y'' ≠ 0}}
| |
| |-
| |
| |align="center"| {{math|''x''<sup>2</sup>}}
| |
| |align="center"| [[square root|{{sqrt|{{mvar|y}}}}]]
| |
| | {{math|''x'', ''y'' ≥ 0}} only
| |
| |-
| |
| |align="center"| [[cube (algebra)|{{math|''x''<sup>3</sup>}}]]
| |
| |align="center"| [[cube root|{{radic|{{mvar|y}}|3}}]]
| |
| | no restriction on {{mvar|x}} and {{mvar|y}}
| |
| |-
| |
| |align="center"| {{math|''x''<sup>''p''</sup>}}
| |
| |align="center"| {{math|''y''<sup>1/''p''</sup>}} (i.e. {{radic|{{mvar|y}}|{{mvar|p}}}})
| |
| | {{math|''x'', ''y'' ≥ 0}} in general, {{math|''p'' ≠ 0}}
| |
| |-
| |
| |align="center"| {{math|[[e (mathematical constant)|''e'']]<sup>''x''</sup>}}
| |
| |align="center"| {{math|[[natural logarithm|ln]] ''y''}}
| |
| | {{math|''y'' > 0}}
| |
| |-
| |
| |align="center"| {{math|''a''<sup>''x''</sup>}}
| |
| |align="center"| {{math|[[logarithm|log]]<sub>''a''</sub> ''y''}}
| |
| | {{math|''y'' > 0}} and {{math|''a'' > 0}}
| |
| |-
| |
| |align="center"| [[trigonometric function]]s
| |
| |align="center"| [[inverse trigonometric functions]]
| |
| | various restrictions (see table below)
| |
| |}
| |
| | |
| ===Formula for the inverse===
| |
| One approach to finding a formula for {{math|''f''<sup> −1</sup>}}, if it exists, is to solve the [[equation]] {{math|1= ''y'' = ''f''(''x'') }} for {{mvar|x}}. For example, if {{mvar|f}} is the function
| |
| | |
| :<math>f(x) = (2x + 8)^3 </math>
| |
| | |
| then we must solve the equation {{math|1= ''y'' = (2''x'' + 8)<sup>3</sup>}} for {{mvar|x}}:
| |
| | |
| :<math>\begin{align}
| |
| y & = (2x+8)^3 \\
| |
| \sqrt[3]{y} & = 2x + 8 \\
| |
| \sqrt[3]{y} - 8 & = 2x \\
| |
| \dfrac{\sqrt[3]{y} - 8}{2} & = x .
| |
| \end{align}</math>
| |
| | |
| Thus the inverse function {{math|''f''<sup> −1</sup>}} is given by the formula
| |
| | |
| :<math>f^{-1}(y) = \dfrac{\sqrt[3]{y} - 8}{2} .</math>
| |
| | |
| Sometimes the inverse of a function cannot be expressed by a formula with a finite number of terms. For example, if {{mvar|f}} is the function
| |
| | |
| :<math>f(x) = x - \sin x ,</math>
| |
| | |
| then {{mvar|f}} is one-to-one, and therefore possesses an inverse function {{math|''f''<sup> −1</sup>}}. The [[Kepler's Equation#Inverse Kepler equation|formula for this inverse]] has an infinite number of terms:<br>
| |
| :<math> f^{-1}(y) =
| |
| \displaystyle \sum_{n=1}^{\infty}
| |
| {\frac{y^{\frac{n}{3}}}{n!}} \lim_{ \theta \to 0} \left(
| |
| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d} \theta^{\,n-1}} \left(
| |
| \frac{ \theta }{ \sqrt[3]{ \theta - \sin( \theta )} } ^n \right)
| |
| \right)
| |
| </math>
| |
| | |
| ===Graph of the inverse===
| |
| [[Image:Inverse Function Graph.png|thumb|right|The graphs of {{math|1= ''y'' = ''f''(''x'') }} and {{math|1= ''y'' = ''f''<sup> −1</sup>(''x'')}}. The dotted line is {{math|1= ''y'' = ''x''}}.]]
| |
| If {{mvar|f}} is invertible, then the graph of the function
| |
| | |
| :<math>y = f^{-1}(x)</math>
| |
| | |
| is the same as the graph of the equation
| |
| | |
| :<math>x = f(y) .</math>
| |
| | |
| This is identical to the equation {{math|1= ''y'' = ''f''(''x'') }} that defines the graph of {{mvar|f}}, except that the roles of {{mvar|x}} and {{mvar|y}} have been reversed. Thus the graph of {{math|''f''<sup> −1</sup>}} can be obtained from the graph of {{mvar|f}} by switching the positions of the {{mvar|x}} and {{mvar|y}} axes. This is equivalent to [[Reflection (mathematics)|reflecting]] the graph across the line
| |
| {{math|1= ''y'' = ''x''}}.
| |
| | |
| ===Inverses and derivatives===
| |
| A [[continuous function]] {{mvar|f}} is one-to-one (and hence invertible) if and only if it is either strictly [[monotonic function|increasing or decreasing]] (with no local [[maxima and minima|maxima or minima]]). For example, the function
| |
| | |
| :<math>f(x) = x^3 + x</math>
| |
| | |
| is invertible, since the [[derivative]]
| |
| {{math|1= ''f′''(''x'') = 3''x''<sup>2</sup> + 1 }} is always positive.
| |
| | |
| If the function {{mvar|f}} is [[differentiable]], then the inverse {{math|''f''<sup> −1</sup>}} will be differentiable as long as {{math| ''f′''(''x'') ≠ 0}}. The derivative of the inverse is given by the [[inverse function theorem]]:
| |
| :<math>\left(f^{-1}\right)^\prime (y) = \frac{1}{f'\left(f^{-1}(y)\right)} . </math>
| |
| If we set {{math|1= ''x'' = ''f''<sup> −1</sup>(''y'')}}, then the formula above can be written
| |
| :<math>\frac{dx}{dy} = \frac{1}{dy / dx} . </math>
| |
| This result follows from the [[chain rule]] (see the article on [[inverse functions and differentiation]]).
| |
| | |
| The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable [[real multivariable function|multivariable function]] {{math| ''f '': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is invertible in a neighborhood of a point {{mvar|p}} as long as the [[Jacobian matrix and determinant|Jacobian matrix]] of {{mvar|f}} at {{mvar|p}} is [[invertible matrix|invertible]]. In this case, the Jacobian of {{math|''f''<sup> −1</sup>}} at {{math|''f''(''p'')}} is the [[matrix inverse]] of the Jacobian of {{mvar|f}} at {{mvar|p}}.
| |
| | |
| == Real-world examples ==
| |
| 1. Let {{mvar|f}} be the function that converts a temperature in degrees [[Celsius]] to a temperature in degrees [[Fahrenheit]]:
| |
| :<math> F = f(C) = \tfrac95 C + 32 ;</math>
| |
| then its inverse function converts degrees Fahrenheit to degrees Celsius:
| |
| :<math> C = f^{-1}(F) = \tfrac59 (F - 32) ,</math>
| |
| since
| |
| :<math> f^{-1}\left( \, f(C) \, \right) = f^{-1}\left( \, \tfrac95 C + 32 \, \right) = \tfrac59 \left( \left( \, \tfrac95 C + 32 \, \right) - 32 \right) = C\text{, for every }C\text{.} </math>
| |
| | |
| 2. Suppose {{mvar|f}} 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 twins (or triplets) 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,
| |
| :<math>\begin{align}
| |
| f(\text{Allan})&=2005 , \quad & f(\text{Brad})&=2007 , \quad & f(\text{Cary})&=2001 \\
| |
| f^{-1}(2005)&=\text{Allan} , \quad & f^{-1}(2007)&=\text{Brad} , \quad & f^{-1}(2001)&=\text{Cary}
| |
| \end{align}
| |
| </math>
| |
| | |
| 3. Let {{mvar|R}} be the function that leads to an {{mvar|x}} percentage rise of some quantity, and {{mvar|F}} be the function producing an {{mvar|x}} percentage fall. Applied to $100 with {{mvar|x}} = 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.
| |
| | |
| ==Generalizations==
| |
| ===Partial inverses===
| |
| [[Image:Inverse Square Graph.png|thumb|right|The square root of {{mvar|x}} is a partial inverse to {{math|1= ''f''(''x'') = ''x''<sup>2</sup>}}.]]
| |
| Even if a function {{mvar|f}} is not one-to-one, it may be possible to define a '''partial inverse''' of {{mvar|f}} by [[Function (mathematics)#Restrictions and extensions|restricting]] the domain. For example, the function
| |
| | |
| :<math>f(x) = x^2</math>
| |
| | |
| is not one-to-one, since {{math|1= ''x''<sup>2</sup> = (−''x'')<sup>2</sup>}}. However, the function becomes one-to-one if we restrict to the domain {{math| ''x'' ≥ 0}}, in which case
| |
| | |
| :<math>f^{-1}(y) = \sqrt{y} . </math>
| |
| | |
| (If we instead restrict to the domain {{math| ''x'' ≤ 0}}, then the inverse is the negative of the square root of {{mvar|y}}.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a [[multivalued function]]:
| |
| | |
| :<math>f^{-1}(y) = \pm\sqrt{y} . </math> | |
| | |
| [[File:Inversa d'una cúbica gràfica.png|right|thumb|The inverse of this [[cubic function]] has three branches.]]
| |
| Sometimes this multivalued inverse is called the '''full inverse''' of {{mvar|f}}, and the portions (such as {{sqrt|{{mvar|x}}}} and −{{sqrt|{{mvar|x}}}}) are called ''branches''. The most important branch of a multivalued function (e.g. the positive square root) is called the ''[[principal branch]]'', and its value at {{mvar|y}} is called the ''principal value'' of {{math|''f''<sup> −1</sup>(''y'')}}.
| |
| | |
| For a continuous function on the real line, one branch is required between each pair of [[minima and maxima|local extrema]]. For example, the inverse of a [[cubic function]] with a local maximum and a local minimum has three branches (see the picture to the right).
| |
| | |
| [[Image:Gràfica_del_arcsinus.png|right|thumb|The [[arcsine]] is a partial inverse of the [[sine]] function.]]
| |
| These considerations are particularly important for defining the inverses of [[trigonometric functions]]. For example, the [[sine function]] is not one-to-one, since
| |
| | |
| :<math>\sin(x + 2\pi) = \sin(x)</math>
| |
| | |
| for every real {{mvar|x}} (and more generally {{math|1= sin(''x'' + 2π''n'') = sin(''x'')}} for every [[integer]] {{mvar|n}}). However, the sine is one-to-one on the interval
| |
| {{closed-closed|−{{sfrac|π|2}}, {{sfrac|π|2}}}}, and the corresponding partial inverse is called the [[arcsine]]. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −{{sfrac|π|2}} and {{sfrac|π|2}}. The following table describes the principal branch of each inverse trigonometric function:
| |
| :{| class="wikitable" style="text-align:center"
| |
| |-
| |
| !function
| |
| !Range of usual [[principal value]]
| |
| |-
| |
| | sin<sup>−1</sup> || {{math|−{{sfrac|π|2}} ≤ sin<sup>−1</sup>(''x'') ≤ {{sfrac|π|2}}}}
| |
| |-
| |
| | cos<sup>−1</sup> || {{math|0 ≤ cos<sup>−1</sup>(''x'') ≤ π}}
| |
| |-
| |
| | tan<sup>−1</sup> || {{math|−{{sfrac|π|2}} < tan<sup>−1</sup>(''x'') < {{sfrac|π|2}}}}
| |
| |-
| |
| | cot<sup>−1</sup> || {{math|0 < cot<sup>−1</sup>(''x'') < π}}
| |
| |-
| |
| | sec<sup>−1</sup> || {{math|0 ≤ sec<sup>−1</sup>(''x'') ≤ π}}
| |
| |-
| |
| | csc<sup>−1</sup> || {{math|−{{sfrac|π|2}} ≤ csc<sup>−1</sup>(''x'') ≤ {{sfrac|π|2}}}}
| |
| |-
| |
| |}
| |
| | |
| ===Left and right inverses=== | |
| If {{math|''f'': ''X'' → ''Y''}}, a '''left inverse''' for {{mvar|f}} (or ''[[retract (category theory)|retraction]]'' of {{mvar|f}}) is a function {{math| ''g'': ''Y'' → ''X''}} such that
| |
| | |
| :<math>g \circ f = \mathrm{id}_X . </math>
| |
| | |
| That is, the function {{mvar|g}} satisfies the rule
| |
| | |
| :If <math>\displaystyle f(x) = y</math>, then <math>\displaystyle g(y) = x .</math>
| |
| | |
| Thus, {{mvar|g}} must equal the inverse of {{mvar|f}} on the range of {{mvar|f}}, but may take any values for elements of {{mvar|Y}} not in the range. A function {{mvar|f}} with a left inverse is necessarily injective. In classical mathematics, every injective function {{mvar|f}} necessarily has a left inverse; however, this may fail in [[constructive mathematics]]. For instance, a left inverse of the inclusion {{math|{0,1} → '''R'''}} of the two-element set in the reals violates [[indecomposability]] by giving a [[Retract (category theory)|retraction]] of the real line to the set {{math|{0,1} }}.
| |
| | |
| A '''right inverse''' for {{mvar|f}} (or ''[[section (category theory)|section]]'' of {{mvar|f}}) is a function {{math| ''h'': ''Y'' → ''X''}} such that
| |
| | |
| :<math>f \circ h = \mathrm{id}_Y . </math>
| |
| | |
| That is, the function {{mvar|h}} satisfies the rule
| |
| | |
| :If <math>\displaystyle h(y) = x</math>, then <math>\displaystyle f(x) = y . </math>
| |
| | |
| Thus, {{math|''h''(''y'')}} may be any of the elements of {{mvar|X}} that map to {{mvar|y}} under {{mvar|f}}. A function {{mvar|f}} has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the [[axiom of choice]]).
| |
| | |
| An inverse which is both a left and right inverse must be unique. Likewise, if {{mvar|g}} is a left inverse for {{mvar|f}}, then {{mvar|g}} may or may not be a right inverse for {{mvar|f}}; and if {{mvar|g}} is a right inverse for {{mvar|f}}, then {{mvar|g}} is not necessarily a left inverse for {{mvar|f}}. For example let {{math|''f'': '''R''' → {{closed-open|0, ∞}}}} denote the squaring map, such that {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} for all {{mvar|x}} in {{math|'''R'''}}, and let {{math|{{mvar|g}}: {{closed-open|0, ∞}} → '''R'''}} denote the square root map, such that {{math|''g''(''x'') {{=}} }}{{radic|{{mvar|x}}}} for all {{math|''x'' ≥ 0}}. Then {{math|1=''f''(''g''(''x'')) = ''x''}} for all {{mvar|x}} in {{closed-open|0, ∞}}; that is, {{mvar|g}} is a right inverse to {{mvar|f}}. However, {{mvar|g}} is not a left inverse to {{mvar|f}}, since, e.g., {{math|1=''g''(''f''(−1)) = 1 ≠ −1}}.
| |
| | |
| ===Preimages===
| |
| If {{math|''f'': ''X'' → ''Y''}} is any function (not necessarily invertible), the '''preimage''' (or '''inverse image''') of an element {{math| ''y'' ∈ ''Y''}} is the set of all elements of {{mvar|X}} that map to {{mvar|y}}:
| |
| | |
| :<math>f^{-1}(y) = \left\{ x\in X : f(x) = y \right\} . </math>
| |
| | |
| The preimage of {{mvar|y}} can be thought of as the [[image (mathematics)|image]] of {{mvar|y}} under the (multivalued) full inverse of the function {{mvar|f}}.
| |
| | |
| Similarly, if {{mvar|S}} is any [[subset]] of {{mvar|Y}}, the preimage of {{mvar|S}} is the set of all elements of {{mvar|X}} that map to {{mvar|S}}:
| |
| | |
| :<math>f^{-1}(S) = \left\{ x\in X : f(x) \in S \right\} . </math>
| |
| | |
| For example, take a function {{math|''f'': '''R''' → '''R'''}}, where {{math|''f'': ''x'' ↦ ''x''<sup>2</sup>}}. This function is not invertible for reasons discussed [[#Example: squaring and square root functions|above]]. Yet preimages may be defined for subsets of the codomain:
| |
| | |
| :<math>f^{-1}(\left\{1,4,9,16\right\}) = \left\{-4,-3,-2,-1,1,2,3,4\right\}</math>
| |
| | |
| The preimage of a single element {{math| ''y'' ∈ ''Y''}} – a [[singleton set]] {{math|{''y''} }} – is sometimes called the ''[[fiber (mathematics)|fiber]]'' of {{mvar|y}}. When {{mvar|Y}} is the set of real numbers, it is common to refer to {{math|''f''<sup> −1</sup>(''y'')}} as a ''[[level set]]''.
| |
| | |
| ==See also==
| |
| * [[Inverse functions and differentiation]]
| |
| * [[Inverse function theorem]]
| |
| * [[Inverse relation]]
| |
| | |
| ==References==
| |
| * {{Citation
| |
| | last = Spivak
| |
| | first = Michael
| |
| | date = 1994
| |
| | title = Calculus
| |
| | publisher = Publish or Perish
| |
| | edition = 3rd
| |
| | isbn = 0-914098-89-6
| |
| }}
| |
| * {{Citation
| |
| | last = Stewart
| |
| | first = James
| |
| | date = 2002
| |
| | title = Calculus
| |
| | publisher = Brooks Cole
| |
| | edition = 5th
| |
| | isbn = 978-0-534-39339-7
| |
| }}
| |
| * {{springer|title=Inverse function|id=p/i052360}}
| |
| * [http://en.wikibooks.org/wiki/Algebra/Functions#Inverse_function Wikibook: Functions]
| |
| * [http://www.encyclopediaofmath.org/index.php/Inverse_function Euro Math Society: Inverse Functions]
| |
| * [http://mathworld.wolfram.com/InverseFunction.html Wolfram Mathworld: Inverse Function]
| |
| * [http://education-portal.com/academy/lesson/understanding-and-graphing-the-inverse-function.html Basic outline.]
| |
| | |
| [[Category:Basic concepts in set theory]]
| |
| [[Category:Inverse functions]]
| |