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Supremacy Clause
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The Supremacy Clause of the United States Constitution (Article VI, Clause 2) establishes that the Constitution, federal laws made pursuant to it, and treaties made under its authority, constitute the supreme law of the land.[1] It provides that state courts are bound by the supreme law; in case of conflict between federal and state law, the federal law must be applied. Even state constitutions are subordinate to federal law.[2] In essence, it is a conflict-of-laws rule specifying that certain federal acts take priority over any state acts that conflict with federal law. In this respect, the Supremacy Clause follows the lead of Article XIII of the Articles of Confederation, which provided that "Every State shall abide by the determination of the United States in Congress Assembled, on all questions which by this confederation are submitted to them."[3] A constitutional provision announcing the supremacy of federal law, the Supremacy Clause assumes the underlying priority of federal authority, at least when that authority is expressed in the Constitution itself.[4] No matter what the federal government or the states might wish to do, they have to stay within the boundaries of the Constitution. This makes the Supremacy Clause the cornerstone of the whole American political structure.[5][6]
Contents
[hide]
1 Text
2 Background
2.1 The Federalist Papers
3 Preemption doctrine
4 Supreme Court interpretations
5 See also
6 References
Text[edit]
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
Background[edit]
The Federalist Papers[edit]
In Federalist No. 44, James Madison defends the Supremacy Clause as vital to the functioning of the nation. He noted that state legislatures were invested with all powers not specifically defined in the Constitution, but also said that having the federal government subservient to various state constitutions would be an inversion of the principles of government, concluding that if supremacy were not established "it would have seen the authority of the whole society everywhere subordinate to the authority of the parts; it would have seen a monster, in which the head was under the direction of the members".
Preemption doctrine[edit]
The constitutional principle derived from the Supremacy Clause is federal preemption. Preemption applies regardless of whether the conflicting laws come from legislatures, courts, administrative agencies, or constitutions. For example, the Voting Rights Act of 1965, an act of Congress, preempts state constitutions, and Food and Drug Administration regulations may preempt state court judgments in cases involving prescription drugs.
Congress has preempted state regulation in many areas. In some cases, such as the 1976 Medical Device Regulation Act, Congress preempted all state regulation. In others, such as labels on prescription drugs, Congress allowed federal regulatory agencies to set federal minimum standards, but did not preempt state regulations imposing more stringent standards than those imposed by federal regulators. Where rules or regulations do not clearly state whether or not preemption should apply, the Supreme Court tries to follow lawmakers’ intent, and prefers interpretations that avoid preempting state laws.[7]
Supreme Court interpretations[edit]
In Ware v. Hylton, 3 U.S. (3 Dall.) 199 (1796), the United States Supreme Court for the first time applied the Supremacy Clause to strike down a state statute. Virginia had passed a statute during the Revolutionary War allowing the state to confiscate debt payments by Virginia citizens to British creditors. The Supreme Court found that this Virginia statute was inconsistent with the Treaty of Paris with Britain, which protected the rights of British creditors. Relying on the Supremacy Clause, the Supreme Court held that the treaty superseded Virginia's statute, and that it was the duty of the courts to declare Virginia's statute "null and void".
In Marbury v. Madison, 5 U.S. 137 (1803), the Supreme Court held that Congress cannot pass laws that are contrary to the Constitution, and it is the role of the Judicial system to interpret what the Constitution permits. Citing the Supremacy Clause, the Court found Section 13 of the Judiciary Act of 1789 to be unconstitutional to the extent it purported to enlarge the original jurisdiction of the Supreme Court beyond that permitted by the Constitution.
In Martin v. Hunter's Lessee, 14 U.S. 304 (1816), and Cohens v. Virginia, 19 U.S. 264 (1821), the Supreme Court held that the Supremacy Clause and the judicial power granted in Article III give the Supreme Court the ultimate power to review state court decisions involving issues arising under the Constitution and laws of the United States. Therefore, the Supreme Court has the final say in matters involving federal law, including constitutional interpretation, and can overrule decisions by state courts.
In McCulloch v. Maryland, 17 U.S. (4 Wheat.) 316 (1819), the Supreme Court reviewed a tax levied by Maryland on the federally incorporated Bank of the United States. The Court found that if a state had the power to tax a federally incorporated institution, then the state effectively had the power to destroy the federal institution, thereby thwarting the intent and purpose of Congress. This would make the states superior to the federal government. The Court found that this would be inconsistent with the Supremacy Clause, which makes federal law superior to state law. The Court therefore held that Maryland's tax on the bank was unconstitutional because the tax violated the Supremacy Clause.
In Ableman v. Booth, 62 U.S. 506 (1859), the Supreme Court held that state courts cannot issue rulings that contradict the decisions of federal courts, citing the Supremacy Clause, and overturning a decision by the Supreme Court of Wisconsin. Specifically, the court found it was illegal for state officials to interfere with the work of U.S. Marshals enforcing the Fugitive Slave Act or to order the release of federal prisoners held for violation of that Act. The Supreme Court reasoned that because the Supremacy Clause established federal law as the law of the land, the Wisconsin courts could not nullify the judgments of a federal court. The Supreme Court held that under Article III of the Constitution, the federal courts have the final jurisdiction in all cases involving the Constitution and laws of the United States, and that the states therefore cannot interfere with federal court judgments.
In Pennsylvania v. Nelson, 350 U.S. 497 (1956) the Supreme Court struck down the Pennsylvania Sedition Act, which made advocating the forceful overthrow of the federal government a crime under Pennsylvania state law. The Supreme Court held that when federal interest in an area of law is sufficiently dominant, federal law must be assumed to preclude enforcement of state laws on the same subject; and a state law is not to be declared a help when state law goes farther than Congress has seen fit to go.
In Reid v. Covert, 354 U.S. 1 (1957), the Supreme Court held that international treaties and laws made pursuant to them must comply with the Constitution.
In Cooper v. Aaron, 358 U.S. 1 (1958), the Supreme Court rejected attempts by Arkansas to nullify the Court's school desegregation decision, Brown v. Board of Education. The state of Arkansas, acting on a theory of states' rights, had adopted several statutes designed to nullify the desegregation ruling. The Supreme Court relied on the Supremacy Clause to hold that the federal law controlled and could not be nullified by state statutes or officials.
In Edgar v. MITE Corp., 457 U.S. 624 (1982), the Supreme Court ruled: "A state statute is void to the extent that it actually conflicts with a valid Federal statute". In effect, this means that a State law will be found to violate the Supremacy Clause when either of the following two conditions (or both) exist:[8]
Compliance with both the Federal and State laws is impossible
"State law stands as an obstacle to the accomplishment and execution of the full purposes and objectives of Congress"
In 1920, the Supreme Court applied the Supremacy Clause to international treaties, holding in the case of Missouri v. Holland, 252 U.S. 416, that the Federal government's ability to make treaties is supreme over any state concerns that such treaties might abrogate states' rights arising under the Tenth Amendment.
The Supreme Court has also held that only specific, "unmistakable" acts of Congress may be held to trigger the Supremacy Clause. Montana had imposed a 30 percent tax on most sub-bituminous coal mined there. The Commonwealth Edison Company and other utility companies argued, in part, that the Montana tax "frustrated" the broad goals of the federal energy policy. However, in the case of Commonwealth Edison Co. v. Montana, 453 U.S. 609 (1981), the Supreme Court disagreed. Any appeal to claims about "national policy", the Court said, were insufficient to overturn a state law under the Supremacy Clause unless "the nature of the regulated subject matter permits no other conclusion, or that the Congress has unmistakably so ordained".[9]
However, in the case of California v. ARC America Corp., 490 U.S. 93 (1989), the Supreme Court held that if Congress expressedly intended to act in an area, this would trigger the enforcement of the Supremacy Clause, and hence nullify the state action. The Supreme Court further found in Crosby v. National Foreign Trade Council, 530 U.S. 363 (2000), that even when a state law is not in direct conflict with a federal law, the state law could still be found unconstitutional under the Supremacy Clause if the "state law is an obstacle to the accomplishment and execution of Congress's full purposes and objectives".[10] Congress need not expressly assert any preemption over state laws either, because Congress may implicitly assume this preemption under the Constitution.[11]
See also[edit]
Federal preemption
Intergovernmental immunity
Interposition
Necessary and Proper Clause
States' rights
Section 109 of the Constitution of Australia – analogous section of the Constitution of Australia
Paramountcy (Canada) – analogous doctrine in Canadian constitutional law
Primacy of European Union law – analogous doctrine in European Union law
References[edit]
Jump up ^ Cornell University Law School. "Supremacy Clause". law.cornell.edu.
Jump up ^ Burnham, William (2006). Introduction to the Law and Legal System of the United States, 4th ed. St. Paul: Thomson West. p. 41.
Jump up ^ Lawson, Gary. "Essays on Article V: Supremacy Clause". The Heritage Foundation. Retrieved March 23, 2016.
Jump up ^ Morrison, Alan B. (1998). "Preemption Controversies". Fundamentals of American law. Oxford University Press US. p. 31. ISBN 978-0-19-876405-2.
Jump up ^ Skousen, W. Cleon (1985). The Making of America – The Substance and Meaning of the Constitution. Washington D.C.: National Center for Constitutional Studies. p. 657.
Jump up ^ Drahozal, Christopher R. (2004). The Supremacy Clause: A Reference Guide to the United States Constitution. Praeger. p. xiv.
Jump up ^ Cornell University Law School. "Preemption". law.cornell.edu.
Jump up ^ Dow Chemical Co. v. Exxon Corp., 139 F.3d 1470 (Fed Cir 1998).
Jump up ^ Commonwealth Edison Co. v. Montana, 453 U.S. 609, 634, quoting Florida Lime & Avocado Growers, Inc. v. Paul, 373 U.S. 132, 142 (1963).
Jump up ^ Crosby v. National Foreign Trade Council, 530 U.S. 363, 372-374.
Jump up ^ Crosby v. National Foreign Trade Council, 530 U.S. 363, 386-388.
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San Jose del Monte National Trade School
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San Jose del Monte National Trade School
SJDMNTS
Location
San Jose del Monte City, Bulacan, Philippines 3023
Philippines
Information
School type
Vocational School
Established
1977
School number
301061
Head of school
Alex D. Angeles
Head teacher
Jeaz Campano
Language
English, Tagalog
Hours in school day
9hour's
Song
"SJDMNTS Hymn"
Newspaper
Lunduyan and The Settlers
San Jose del Monte National Trade School is a public secondary school in San Jose del Monte City, Bulacan, Philippines. The head of the school is Mr. Alex D. Angeles. [1]
References[edit]
Jump up ^ "Masterlist of Public Schools". Department of Education of the Philippines - DepEd. Retrieved 2010-04-28. School ID 301061
Philippines portal
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14°50′53″N 121°3′1″E / 14.84806°N 121.05028°E / 14.84806; 121.05028Coordinates: 14°50′53″N 121°3′1″E / 14.84806°N 121.05028°E / 14.84806; 121.05028

This Philippines school-related article is a stub. You can help Wikipedia by expanding it.
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In mathematics, an inverse function (or anti-function[1]) is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. I.e., f(x) = y if and only if g(y) = x.[2][3]
As a simple example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case that means that we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by,
g ( y ) = y + 7 5 . {\displaystyle g(y)={\frac {y+7}{5}}.}
With y = 5x − 7 we have that f(x) = y and g(y) = x.
Not all functions have inverse functions. In order for a function f: X → Y to have an inverse,[nb 1] it must have the property that for every y in Y there must be one, and only one x in X so that f(x) = y. This property ensures that a function g: Y → X will exist having the necessary relationship with f.
Contents
[hide]
1 Definitions
1.1 Example: Squaring and square root functions
1.2 Inverses and composition
1.3 Note on notation
2 Properties
2.1 Uniqueness
2.2 Symmetry
2.3 Self-inverses
3 Inverses in calculus
3.1 Formula for the inverse
3.2 Graph of the inverse
3.3 Inverses and derivatives
4 Real-world examples
5 Generalizations
5.1 Partial inverses
5.2 Left and right inverses
5.3 Preimages
6 See also
7 Notes
8 References
9 Further reading
10 External links
Definitions[edit]
If f maps X to Y, then f −1 maps Y back to X.
Let f be a function whose domain is the set X, and whose image (range) is the set Y. Then f is invertible if there exists a function g with domain Y and image X, with the property:
f ( x ) = y ⇔ g ( y ) = x . {\displaystyle f(x)=y\,\,\Leftrightarrow \,\,g(y)=x.}
If f is invertible, the function g is unique,[4] which means that there is exactly one function g satisfying this property (no more, no less). That function g is then called the inverse of f, and is usually denoted as f −1.[nb 2]
Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function.[5]
Not all functions have an inverse. For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X; a function f with this property is called one-to-one or an injection. If f −1 is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. Functions with this property are called surjections. This property is satisfied by definition if Y is the image (range) of f, but may not hold in a more general context. To be invertible a function must be both an injection and a surjection. Such functions are called bijections. The inverse of an injection f: X → Y that is not a bijection, that is, a function that is not a surjection, is only a partial function on Y, which means that for some y ∈ Y, f −1(y) is undefined. If a function f is invertible, then both it and its inverse function f−1 are bijections.
There is another convention used in the definition of functions. This can be referred to as the "set-theoretic" or "graph" definition using ordered pairs in which a codomain is never referred to.[6] Under this convention all functions are surjections,[nb 3] and so, being a bijection simply means being an injection. Authors using this convention may use the phrasing that a function is invertible if and only if it is an injection.[7] The two conventions need not cause confusion as long as it is remembered that in this alternate convention the codomain of a function is always taken to be the range of the function.
Example: Squaring and square root functions[edit]
The function f:ℝ → [0,∞) given by f(x) = x2 is not injective since each possible result y (except 0) corresponds to two different starting points in X – one positive and one negative, and so this function is not invertible. With this type of function it is impossible to deduce an input from its output. Such a function is called non-injective or, in some applications, information-losing.[citation needed]
If the domain of the function is restricted to the nonnegative reals, that is, the function is redefined to be f: [0, ∞) → [0, ∞) with the same rule as before, then the function is bijective and so, invertible.[8] The inverse function here is called the (positive) square root function.
Inverses and composition[edit]
See also: Inverse element
If f is an invertible function with domain X and range Y, then
f − 1 ( f ( x ) ) = x {\displaystyle f^{-1}\left(\,f(x)\,\right)=x} , for every x ∈ X . {\displaystyle x\in X.}
Using the composition of functions we can rewrite this statement as follows:
f − 1 ∘ f = id X , {\displaystyle f^{-1}\circ f=\operatorname {id} _{X},}
where idX is the identity function on the set X; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism.
Considering function composition helps to understand the notation f −1. Repeatedly composing a function with itself is called iteration. If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) = f (f (x)), etc. Since f −1(f (x)) = x, composing f −1 and f n yields f n−1, "undoing" the effect of one application of f.
Note on notation[edit]
While the notation f −1(x) might be misunderstood, (f(x))−1 certainly denotes the multiplicative inverse of f(x) and has nothing to do with the inverse function of f.
In keeping with the general notation, some English[1] authors use expressions like sin−1(x) to denote the inverse of the sine function applied to x (actually a partial inverse; see below)[9] Other authors feel that this may be confused with the notation for the multiplicative inverse of sin (x), which can be denoted as (sin (x))−1. To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcus).[1][10][11] For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin(x).[1][10][11] Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin area).[11] For instance, the inverse of the hyperbolic sine function is typically written as arsinh(x).[11] Other inverse special functions are sometimes prefixed with the prefix "inv" if the ambiguity of the f −1 notation should be avoided.[1][11]
Properties[edit]
Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of inverse relations.
Uniqueness[edit]
If an inverse function exists for a given function f, then it is unique.[12] This follows since the inverse function must be the inverse relation which is completely determined by f.
Symmetry[edit]
There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and range Y, then its inverse f −1 has domain Y and range X, and the inverse of f −1 is the original function f. In symbols, for functions f:X → Y and f−1:Y → X,[12]
f − 1 ∘ f = id X {\displaystyle f^{-1}\circ f=\operatorname {id} _{X}} and f ∘ f − 1 = id Y . {\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}.}
This statement is a consequence of the implication that for f to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by[13]
( f − 1 ) − 1 = f . {\displaystyle \left(f^{-1}\right)^{-1}=f.}
The inverse of g ∘ f is f −1 ∘ g −1.
The inverse of a composition of functions is given by[14]
( g ∘ f ) − 1 = f − 1 ∘ g − 1 . {\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}.}
Notice that the order of g and f have been reversed; to undo f followed by g, we must first undo g and then undo f.
For example, let f(x) = 3x and let g(x) = x + 5. Then the composition g ∘ f is the function that first multiplies by three and then adds five,
( g ∘ f ) ( x ) = 3 x + 5. {\displaystyle (g\circ f)(x)=3x+5.}
To reverse this process, we must first subtract five, and then divide by three,
( g ∘ f ) − 1 ( x ) = 1 3 ( x − 5 ) . {\displaystyle (g\circ f)^{-1}(x)={\tfrac {1}{3}}(x-5).}
This is the composition (f −1 ∘ g −1)(x).
Self-inverses[edit]
If X is a set, then the identity function on X is its own inverse:
id X − 1 = id X . {\displaystyle {\operatorname {id} _{X}}^{-1}=\operatorname {id} _{X}.}
More generally, a function f : X → X is equal to its own inverse if and only if the composition f ∘ f is equal to idX. Such a function is called an involution.
Inverses in calculus[edit]
Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functions are often defined through formulas, such as:
f ( x ) = ( 2 x + 8 ) 3 . {\displaystyle f(x)=(2x+8)^{3}.}
A surjective function 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 y = f(x) has, for each possible y value only one corresponding x value, and thus passes the horizontal line test.
The following table shows several standard functions and their inverses:
Function f(x)
Inverse f −1(y)
Notes
x + a y − a
a − x a − y
mx y/m m ≠ 0
1/x (i.e. x−1) 1/y (i.e. y−1) x, y ≠ 0
x2 √y (i.e. y1/2) x, y ≥ 0 only
x3 3√y (i.e. y1/3) no restriction on x and y
xp p√y (i.e. y1/p) x, y ≥ 0 if p is even; integer p > 0
2x lb y y > 0
ex ln y y > 0
10x log y y > 0
ax loga y y > 0 and a > 0
trigonometric functions inverse trigonometric functions various restrictions (see table below)
hyperbolic functions inverse hyperbolic functions various restrictions
Formula for the inverse[edit]
One approach to finding a formula for f −1, if it exists, is to solve the equation y = f(x) for x.[15] For example, if f is the function
f ( x ) = ( 2 x + 8 ) 3 {\displaystyle f(x)=(2x+8)^{3}}
then we must solve the equation y = (2x + 8)3 for x:
y = ( 2 x + 8 ) 3 y 3 = 2 x + 8 y 3 − 8 = 2 x y 3 − 8 2 = x . {\displaystyle {\begin{aligned}y&=(2x+8)^{3}\\{\sqrt[{3}]{y}}&=2x+8\\{\sqrt[{3}]{y}}-8&=2x\\{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&=x.\end{aligned}}}
Thus the inverse function f −1 is given by the formula
f − 1 ( y ) = y 3 − 8 2 . {\displaystyle f^{-1}(y)={\frac {{\sqrt[{3}]{y}}-8}{2}}.}
Sometimes the inverse of a function cannot be expressed by a formula with a finite number of terms. For example, if f is the function
f ( x ) = x − sin  x , {\displaystyle f(x)=x-\sin x,}
then f is a bijection, and therefore possesses an inverse function f −1. The formula for this inverse has an infinite number of terms:
f − 1 ( y ) = ∑ n = 1 ∞ y n / 3 n ! lim θ → 0 ( d n − 1 d θ n − 1 ( θ θ − sin  ( θ ) 3 ) n ) . {\displaystyle f^{-1}(y)=\sum _{n=1}^{\infty }{\frac {y^{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 )}}}\right)^{n}\right).}
Graph of the inverse[edit]
The graphs of y = f(x) and y = f −1(x). The dotted line is y = x.
If f is invertible, then the graph of the function
y = f − 1 ( x ) {\displaystyle y=f^{-1}(x)}
is the same as the graph of the equation
x = f ( y ) . {\displaystyle x=f(y).}
This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed. Thus the graph of f −1 can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line y = x.[16]
Inverses and derivatives[edit]
A continuous function f is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function
f ( x ) = x 3 + x {\displaystyle f(x)=x^{3}+x}
is invertible, since the derivative f′(x) = 3x2 + 1 is always positive.
If the function f is differentiable on an interval I and f′(x) ≠ 0 for each x ∈ I, then the inverse f −1 will be differentiable on f(I).[17] If y = f(x), the derivative of the inverse is given by the inverse function theorem,
( f − 1 ) ′ ( y ) = 1 f ′ ( x ) . {\displaystyle \left(f^{-1}\right)^{\prime }(y)={\frac {1}{f'\left(x\right)}}.}
Using Leibniz's notation the formula above can be written as
d x d y = 1 d y / d x . {\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}
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 multivariable function f : Rn → Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible. In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the Jacobian of f at p.
Real-world examples[edit]
Let f be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit,
F = f ( C ) = 9 5 C + 32 ; {\displaystyle F=f(C)={\tfrac {9}{5}}C+32;}
then its inverse function converts degrees Fahrenheit to degrees Celsius,
C = f − 1 ( F ) = 5 9 ( F − 32 ) , {\displaystyle C=f^{-1}(F)={\tfrac {5}{9}}(F-32),}
since
f − 1 ( f ( C ) ) = f − 1 ( 9 5 C + 32 ) = 5 9 ( ( 9 5 C + 32 ) − 32 ) = C , for every value of C , and f ( f − 1 ( F ) ) = f ( 5 9 ( F − 32 ) ) = 9 5 ( 5 9 ( F − 32 ) ) + 32 = F , for every value of F . {\displaystyle {\begin{aligned}f^{-1}(f(C))={}&f^{-1}\left({\tfrac {9}{5}}C+32\right)={\tfrac {5}{9}}\left(({\tfrac {9}{5}}C+32)-32\right)=C,\\&{\text{for every value of }}C,{\text{ and }}\\[6pt]f\left(f^{-1}(F)\right)={}&f\left({\tfrac {5}{9}}(F-32)\right)={\tfrac {9}{5}}\left({\tfrac {5}{9}}(F-32)\right)+32=F,\\&{\text{for every value of }}F.\end{aligned}}}
Suppose 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 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,
f ( Allan ) = 2005 , f ( Brad ) = 2007 , f ( Cary ) = 2001 f − 1 ( 2005 ) = Allan , f − 1 ( 2007 ) = Brad , f − 1 ( 2001 ) = Cary {\displaystyle {\begin{aligned}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{aligned}}}
Let R be the function that leads to an x percentage rise of some quantity, and F be the function producing an x percentage fall. Applied to $100 with 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[edit]
Partial inverses[edit]
The square root of x is a partial inverse to f(x) = x2.
Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function
f ( x ) = x 2 {\displaystyle f(x)=x^{2}}
is not one-to-one, since x2 = (−x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case
f − 1 ( y ) = y . {\displaystyle f^{-1}(y)={\sqrt {y}}.}
(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:
f − 1 ( y ) = ± y . {\displaystyle f^{-1}(y)=\pm {\sqrt {y}}.}
The inverse of this cubic function has three branches.
Sometimes this multivalued inverse is called the full inverse of f, and the portions (such as √x and −√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 y is called the principal value of f −1(y).
For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture).
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
sin  ( x + 2 π ) = sin  ( x ) {\displaystyle \sin(x+2\pi )=\sin(x)}
for every real x (and more generally sin(x + 2πn) = sin(x) for every integer n). However, the sine is one-to-one on the interval [−π/2, 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 −π/2 and π/2. The following table describes the principal branch of each inverse trigonometric function:[18]
function
Range of usual principal value
arcsin −π/2 ≤ sin−1(x) ≤ π/2
arccos 0 ≤ cos−1(x) ≤ π
arctan −π/2 < tan−1(x) < π/2
arccot 0 < cot−1(x) < π
arcsec 0 ≤ sec−1(x) ≤ π
arccsc −π/2 ≤ csc−1(x) ≤ π/2
Left and right inverses[edit]
If f: X → Y, a left inverse for f (or retraction of f ) is a function g: Y → X such that
g ∘ f = id X . {\displaystyle g\circ f=\operatorname {id} _{X}.}
That is, the function g satisfies the rule
If f ( x ) = y , then g ( y ) = x . {\displaystyle {\text{If }}f(x)=y,{\text{ then }}g(y)=x.}
Thus, g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with a left inverse is necessarily injective. In classical mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1} .
A right inverse for f (or section of f ) is a function h: Y → X such that
f ∘ h = id Y . {\displaystyle f\circ h=\operatorname {id} _{Y}.}
That is, the function h satisfies the rule
If h ( y ) = x {\displaystyle \displaystyle h(y)=x} , then f ( x ) = y . {\displaystyle \displaystyle f(x)=y.}
Thus, h(y) may be any of the elements of X that map to y under f. A function 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. However, if g is a left inverse for f, then g may or may not be a right inverse for f; and if g is a right inverse for f, then g is not necessarily a left inverse for f. For example, let f: R → [0, ∞) denote the squaring map, such that f(x) = x2 for all x in R, and let g: [0, ∞) → R denote the square root map, such that g(x) = √x for all x ≥ 0. Then f(g(x)) = x for all x in [0, ∞); that is, g is a right inverse to f. However, g is not a left inverse to f, since, e.g., g(f(−1)) = 1 ≠ −1.
Preimages[edit]
If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is the set of all elements of X that map to y:
f − 1 ( { y } ) = { x ∈ X : f ( x ) = y } . {\displaystyle f^{-1}(\{y\})=\left\{x\in X:f(x)=y\right\}.}
The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f.
Similarly, if S is any subset of Y, the preimage of S is the set of all elements of X that map to S:
f − 1 ( S ) = { x ∈ X : f ( x ) ∈ S } . {\displaystyle f^{-1}(S)=\left\{x\in X:f(x)\in S\right\}.}
For example, take a function f: R → R, where f: x ↦ x2. This function is not invertible for reasons discussed above. Yet preimages may be defined for subsets of the codomain:
f − 1 ( { 1 , 4 , 9 , 16 } ) = { − 4 , − 3 , − 2 , − 1 , 1 , 2 , 3 , 4 } {\displaystyle f^{-1}(\left\{1,4,9,16\right\})=\left\{-4,-3,-2,-1,1,2,3,4\right\}}
The preimage of a single element y ∈ Y – a singleton set {y} – is sometimes called the fiber of y. When Y is the set of real numbers, it is common to refer to f −1({y}) as a level set.
See also[edit]
Lagrange inversion theorem, gives the Taylor series expansion of the inverse function of an analytic function
Reversible computing
Notes[edit]
Jump up ^ It is a common practice, when no ambiguity can arise, to leave off the term function and just refer to an inverse.
Jump up ^ Not to be confused with numerical exponentiation such as taking the multiplicative inverse of a nonzero real number.
Jump up ^ So this term is never used in this convention.
References[edit]
^ Jump up to: a b c d e Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [14] Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. […] α = arcsin m: It is frequently read "arc-sine m" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] 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 sin m is perhaps better still on account of its general applicability. […]
Jump up ^ Keisler, Howard Jerome. "Differentiation" (PDF). Retrieved 2015-01-24. §2.4
Jump up ^ Scheinerman, Edward R. (2013). Mathematics: A Discrete Introduction. Brooks/Cole. p. 173. ISBN 978-0840049421.
Jump up ^ Devlin 2004, p. 101, Theorem 4.5.1
Jump up ^ Smith, Eggen & St. Andre 2006, p. 202, Theorem 4.9
Jump up ^ Wolf 1998, p.198
Jump up ^ Fletcher & Patty 1988, p. 116 Theorem 5.1
Jump up ^ Lay 2006, p.69 Example 7.24
Jump up ^ Thomas 1972, pp. 304-309
^ Jump up to: a b Korn, Grandino Arthur; Korn, Theresa M. (2000) [1961]. "21.2.-4. Inverse Trigonometric Functions". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review (3 ed.). Mineola, New York, USA: Dover Publications, Inc. p. 811;. ISBN 978-0-486-41147-7.
^ Jump up to: a b c d e Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. An Atlas of Functions: with Equator, the Atlas Function Calculator (2 ed.). Springer Science+Business Media, LLC. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48806-6. LCCN 2008937525.
^ Jump up to: a b Wolf 1998, p. 208 Theorem 7.2
Jump up ^ Smith, Eggen & St. Andre 2006, pg. 141 Theorem 3.3(a)
Jump up ^ Lay 2006, p. 71 Theorem 7.26
Jump up ^ Devlin 2004, p. 101
Jump up ^ Briggs & Cochran 2011, pp. 28-29
Jump up ^ Lay 2006, p. 246 Theorem 26.10
Jump up ^ Briggs & Cochran 2011, pp. 39-42
Briggs, William; Cochran, Lyle (2011). Calculus / Early Transcendentals Single Variable. Addison-Wesley. ISBN 978-0-321-66414-3.
Devlin, Keith J. (2004). Sets, Functions, and Logic / An Introduction to Abstract Mathematics (3rd ed.). Chapman & Hall / CRC Mathematics. ISBN 978-1-58488-449-1.
Fletcher, Peter; Patty, C. Wayne (1988). Foundations of Higher Mathematics. PWS-Kent. ISBN 0-87150-164-3.
Lay, Steven R. (2006). Analysis / With an Introduction to Proof (4 ed.). Pearson / Prentice Hall. ISBN 978-0-13-148101-5.
Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006). A Transition to Advanced Mathematics (6 ed.). Thompson Brooks/Cole. ISBN 978-0-534-39900-9.
Thomas, Jr., George Brinton (1972). Calculus and Analytic Geometry Part 1: Functions of One Variable and Analytic Geometry (Alternate ed.). Addison-Wesley.
Wolf, Robert S. (1998). Proof, Logic, and Conjecture / The Mathematician's Toolbox. W. H. Freeman and Co. ISBN 978-0-7167-3050-7.
Further reading[edit]
Spivak, Michael (1994). Calculus (3 ed.). Publish or Perish. ISBN 0-914098-89-6.
Stewart, James (2002). Calculus (5 ed.). Brooks Cole. ISBN 978-0-534-39339-7.
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Inverse function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Wikibook: Functions
Wolfram Mathworld: Inverse Function
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Function of a complex variable

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Smooth function

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Unique (mathematics)

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Composition of functions

values: {\displaystyle f^{-1}\circ f=\operatorname {id} _{X},}

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Inverse trigonometric function

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Arc (function prefix)

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Ar (function prefix)

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Hyperbolic sine

values: {\displaystyle f^{-1}\circ f=\operatorname {id} _{X}}

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values: {\displaystyle f\circ f^{-1}=\operatorname {id} _{Y}.}

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values: {\displaystyle (g\circ f)^{-1}=f^{-1}\circ g^{-1}.}

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Decadic logarithm

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Trigonometric function

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Inverse hyperbolic function

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values: {\displaystyle {\frac {dx}{dy}}={\frac {1}{dy/dx}}.}

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Real multivariable function

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Matrix inverse

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values: {\displaystyle f^{-1}(y)=\pm {\sqrt {y}}.}

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Minima and maxima

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Sine function

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Retract (category theory)

values: {\displaystyle g\circ f=\operatorname {id} _{X}.}

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values: {\displaystyle {\text{If }}f(x)=y,{\text{ then }}g(y)=x.}

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Constructive mathematics

values: {\displaystyle f\circ h=\operatorname {id} _{Y}.}

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Singleton set

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Edward R. Scheinerman

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Dover Publications, Inc.

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Keith J. Devlin

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CRC Mathematics

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Pearson (publisher)

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Thompson Brooks/Cole

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George Brinton Thomas, Jr.

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W. H. Freeman and Co.

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Brooks Cole

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