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Mathematical concept
mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if
Inverse_function
Inverse functions of sin, cos, tan, etc.
mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the
Inverse trigonometric functions
Inverse_trigonometric_functions
Mathematical functions
mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in
Inverse_hyperbolic_functions
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that
Inverse_function_theorem
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of
Inverse_function_rule
Inverse of the gamma function
In mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gamma function. In other words, y
Inverse_gamma_function
Association of one output to each input
interval I, it has an inverse function, which is a real function with domain f(I) and image I. This is how inverse trigonometric functions are defined in terms
Function_(mathematics)
Quickly growing function
recursive function and is therefore not primitive recursive. Since the function f(n) = A(n, n) considered above grows very rapidly, its inverse function, f−1
Ackermann_function
Basic method for pseudo-random number sampling
from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u {\displaystyle
Inverse_transform_sampling
Statistical function that defines the quantiles of a probability distribution
probability distribution's quantile function is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_function
Function that is its own inverse
mathematics, an involution, involutory function, or self-inverse function is a function f that is its own inverse, f(f(x)) = x for all x in the domain of
Involution_(mathematics)
Set of the values of a function
not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of B {\displaystyle B}
Image_(mathematics)
Number which when multiplied by x equals 1
the function f(x) that maps x to 1 x , {\displaystyle {\tfrac {1}{x}},} is one of the simplest examples of a function which is its own inverse (an involution)
Multiplicative_inverse
Sigmoid shape special function
\end{aligned}}} The inverse of Φ is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as probit
Error_function
On converting relations to functions of several real variables
the implicit function theorem. Inverse function theorem Constant rank theorem: Both the implicit function theorem and the inverse function theorem can
Implicit_function_theorem
Functions of an angle
trigonometric functions has a corresponding inverse function and has an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related
Trigonometric_functions
Mathematical function in economics
In economics, an inverse demand function is the mathematical relationship that expresses price as a function of quantity demanded (it is therefore also
Inverse_demand_function
Mathematical relation consisting of a multi-variable function equal to zero
multivariable functions that are continuously differentiable. A common type of implicit function is an inverse function. Not all functions have a unique inverse function
Implicit_function
Mathematical theorem, used in calculus
mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle f^{-1}}
Integral_of_inverse_functions
Technique in integral evaluation
differentiable and have a continuous inverse. This is guaranteed to hold if φ is continuously differentiable by the inverse function theorem. Alternatively, the
Integration_by_substitution
Hyperbolic analogues of trigonometric functions
trigonometric functions. The inverse hyperbolic functions are: inverse hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") inverse hyperbolic
Hyperbolic_functions
Mathematical function, inverse of an exponential function
logb x = y, so log10 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b. The logarithm base 10
Logarithm
Logarithm to the base of the mathematical constant e
real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: e ln x = x if x ∈ R +
Natural_logarithm
Matrix of partial derivatives of a vector-valued function
of a usual function to vector valued functions of several variables. This generalization includes generalizations of the inverse function theorem and
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Operation on mathematical functions
follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (f ∘ g)−1
Function_composition
Mathematical function such that every output has at least one input
domain. Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection
Surjective_function
Instantaneous rate of change (mathematics)
{d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x)} Inverse trigonometric functions: d d x arcsin ( x ) = 1 1 − x 2 {\displaystyle {\frac
Derivative
1 minus the cosine of an angle
the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions
Versine
One-to-one correspondence
there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f, such that each of the two ways for composing the two functions produces an
Bijection
Mathematical function, denoted exp(x) or e^x
x ⋅ exp y {\displaystyle \exp(x+y)=\exp x\cdot \exp y} . Its inverse function, the natural logarithm, ln {\displaystyle \ln } or log {\displaystyle
Exponential_function
Indefinite integral
antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative
Antiderivative
Mathematical function
5^{2}(2-k^{2})+z^{2}-{}}}\cdots } The inverses of the Jacobi elliptic functions can be defined similarly to the inverse trigonometric functions; if x = sn ( ξ , m )
Jacobi_elliptic_functions
Fundamental trigonometric functions
0 {\displaystyle \sin(2\pi )=0} . Sine's "inverse", called arcsine, can then be described not as a function but a relation (for example, all integer multiples
Sine_and_cosine
Asymmetric sigmoid function
function, and then convert it to the equivalent inverse function using the relationship between the two given above. In this way the inverse function
Gompertz_function
Multivalued function in mathematics
Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. The name "product logarithm" can be understood as follows: since the inverse function of f
Lambert_W_function
Geometric line segment whose endpoints lie on a circular arc
the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones: The inverse function exists as well: θ = 2
Chord_(geometry)
Reversal of the order of elements of a binary relation
holds. A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function. The converse
Converse_relation
Probability that random variable X is less than or equal to x
F(x)=p} . This defines the inverse distribution function or quantile function. Some distributions do not have a unique inverse (for example if f X ( x )
Cumulative distribution function
Cumulative_distribution_function
Analytic function that does not satisfy a polynomial equation
algebraic function. The most familiar transcendental functions are the exponential, trigonometric, and hyperbolic functions, and their inverses, such as
Transcendental_function
Mathematical process of finding the derivative of a trigonometric function
sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation. The diagram at right
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
Mathematical method in calculus
integral of an inverse function f−1(x) when the integral of the function f(x) is known. Indeed, the functions x(y) and y(x) are inverses, and the integral
Integration_by_parts
Mapping involving integration between function spaces
in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform. An integral
Integral_transform
Mathematical functions
by using the binomial series. The inverse function of the lemniscate cosine is the lemniscate arccosine. This function is defined by following expression:
Lemniscate_elliptic_functions
Generalization of additive and multiplicative inverses
More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is
Inverse_element
Formula in calculus
for the quotient rule. Suppose that y = g(x) has an inverse function. Call its inverse function f so that we have x = f(y). There is a formula for the
Chain_rule
Process of calculating the causal factors that produced a set of observations
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating
Inverse_problem
Branch of mathematics
the inverse of integration. The fundamental theorem of calculus states: If a function f is continuous on the interval [a, b] and if F is a function whose
Calculus
S-shaped curve
called the sigmoid function. It is also sometimes called the expit, being the inverse function of the logit. The logistic function finds applications
Logistic_function
Mathematical approximation of a function
complex functions, such as logarithms, fractional powers, and inverse trigonometric functions, a principal branch is understood. The exponential function ex
Taylor_series
Rules for computing derivatives of functions
g)]_{x}=[{\text{D}}f]_{g(x)}\cdot [{\text{D}}g]_{x}.} If the function f {\textstyle f} has an inverse function g {\textstyle g} , meaning that g ( f ( x ) ) = x
Differentiation_rules
Mapping which preserves all topological properties of a given space
or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are
Homeomorphism
Matrix of second derivatives
partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Functions such that f(–x) equals f(x) or –f(x)
is even (but not vice versa). If an odd function is invertible, then its inverse is also odd. If a real function has a domain that is self-symmetric with
Even_and_odd_functions
Mathematical function with no sudden changes
has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between
Continuous_function
Integer
is specified inside the function f, its inverse will yield an inverse image, or preimage, of that subset under the function. Exponentiation to negative
−1
Estimate of the importance of a word in a document
called Inverse Document Frequency (idf), which became a cornerstone of term weighting: The specificity of a term can be quantified as an inverse function of
Tf–idf
Function related to statistics and probability theory
implicitly defined by the value at 0 {\textstyle \mathbf {0} } of the inverse function s n − 1 : E d → Θ {\textstyle s_{n}^{-1}:\mathbb {E} ^{d}\to \Theta
Likelihood_function
Function with a smaller domain
restriction of a continuous function is continuous. For a function to have an inverse, it must be one-to-one. If a function f {\displaystyle f} is not
Restriction_(mathematics)
Generalization of the inverse function theorem
Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping
Nash–Moser_theorem
Order-preserving mathematical function
therefore not one-to-one). A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it
Monotonic_function
Course designed to prepare students for calculus
logarithm, to an arbitrary positive base, Euler presents as the inverse of an exponential function. Then the natural logarithm is obtained by taking as base
Precalculus
Generalized mathematical function
roots, logarithms, and inverse trigonometric functions. To define a single-valued function from a complex multivalued function, one may distinguish one
Multivalued_function
Two-parameter family of continuous probability distributions
inverse gamma distribution differently, as a scaled inverse chi-squared distribution. The inverse gamma distribution's probability density function is
Inverse-gamma_distribution
Class of periodic mathematical functions
Abel discovered elliptic functions by taking the inverse function φ {\displaystyle \varphi } of the elliptic integral function α ( x ) = ∫ 0 x d t ( 1
Elliptic_function
Computing joint values of a kinematic chain from a known end position
In computer animation and robotics, inverse kinematics (IK) is the mathematical process of calculating the variable joint parameters needed to place the
Inverse_kinematics
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Special function defined by an integral
(Theis well function)". Journal of Hydrology. 227 (1–4): 287–291. Bibcode:2000JHyd..227..287B. doi:10.1016/S0022-1694(99)00184-5. "Inverse function of the
Exponential_integral
Function in statistics
data transformations. Mathematically, the logit is the inverse of the standard logistic function σ ( x ) = 1 / ( 1 + e − x ) {\displaystyle \sigma (x)=1/(1+e^{-x})}
Logit
Formula for inverting a Taylor series
the inverse function of an analytic function. Lagrange inversion is a special case of the inverse function theorem. Suppose z is defined as a function of
Lagrange_inversion_theorem
Topics referred to by the same term
In mathematics, inverse mapping theorem may refer to: the inverse function theorem on the existence of local inverses for functions with non-singular
Inverse_mapping_theorem
Operation in mathematical calculus
compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations. Although methods of
Integral
Topics referred to by the same term
sentence Additive inverse, the inverse of a number that, when added to the original number, yields zero Compositional inverse, a function that "reverses"
Inverse
trigonometric functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of exponential functions List
Lists_of_integrals
Method of mathematical integration
of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X axis. The
Lebesgue_integral
Statistical method of dividing data into equal-sized intervals for analysis
distribution function of a random variable is known, the q-quantiles are the application of the quantile function (the inverse function of the cumulative
Quantile
Family of continuous probability distributions
generating functions of the Gaussian and inverse Gaussian distributions are inverse of each other (i.e., the graphs of the two cumulant generating functions are
Inverse_Gaussian_distribution
Mathematical operation
In mathematics, the inverse Laplace transform of a function F {\displaystyle F} is a real function f {\displaystyle f} that is piecewise-continuous,
Inverse_Laplace_transform
Result of repeatedly applying a mathematical function
conjugacy below.) If a function is bijective (and so possesses an inverse function), then negative iterates correspond to function inverses and their compositions
Iterated_function
Conditions for switching order of integration in calculus
was used by Leonhard Euler. More formally, the theorem states that if a function is Lebesgue integrable on a rectangle X × Y {\displaystyle X\times Y}
Fubini's_theorem
Multivariate derivative (mathematics)
scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla
Gradient
Notion in calculus
calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent
Differential_of_a_function
Most widely known generalized inverse of a matrix
In mathematics, and in particular linear algebra, the Moore–Penrose inverse A + {\displaystyle A^{+}} of a matrix A {\displaystyle A} , often called
Moore–Penrose_inverse
Derivative of a function with multiple variables
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held
Partial_derivative
Number that, when added to the original number, yields the additive identity
identity |−x| = |x|). Inverse element Inverse function Involution (mathematics) Monoid Multiplicative inverse Reflection (mathematics) Reflection symmetry
Additive_inverse
Entropy of a process with only two probable values
derivative of negative binary entropy is the logit, whose inverse function is the logistic function, which is the derivative of softplus. Softplus can be
Binary_entropy_function
Theorem in mathematics
proving other general properties of differentiable functions. A special case of this theorem for inverse interpolation of the sine was first described by
Mean_value_theorem
Finding values for variables that make an equation true
inverse functions include the nth root (inverse of xn); the logarithm (inverse of ax); the inverse trigonometric functions; and Lambert's W function (inverse
Equation_solving
Generalization of the concept of directional derivative
applications such as the Nash–Moser inverse function theorem in which the function spaces of interest often consist of smooth functions on a manifold. Whereas higher
Gateaux_derivative
Relationship between derivatives and integrals
of as inverses of each other. The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function f , an
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Study of rates of change
is closely related to the inverse function theorem, which states when a function looks like graphs of invertible functions pasted together. Differential
Differential_calculus
Theorem in algebra mathematics
the tor functor. Nakayama's lemma is used to prove a version of the inverse function theorem in algebraic geometry: Let f : X → Y {\textstyle f:X\to Y}
Nakayama's_lemma
Function that preserves distinctness
line test. Functions with left inverses are always injections. That is, given f : X → Y {\displaystyle f:X\to Y} , if there is a function g : Y → X
Injective_function
Logarithm of a complex number
the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense. There are
Complex_logarithm
involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals. The inverse trigonometric functions are also
List of integrals of inverse trigonometric functions
List_of_integrals_of_inverse_trigonometric_functions
Arithmetic operation
than or equal to 3 have analogous inverses); e.g., in the function 3 y = x {\displaystyle {^{3}}y=x} , the two inverses are the cube super-root of y and
Tetration
Circulation density in a vector field
curl(W + grad(f)) = V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown
Curl_(mathematics)
Mathematical transformation
to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative
Legendre_transformation
Smoothed ramp function
transformation: the derivatives are inverse functions) the derivative of softplus is the logistic function, whose inverse function is the logit, which is the derivative
Softplus
Theorem in calculus relating line and double integrals
curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial
Green's_theorem
Modular function in mathematics
{(A^{2}-3)^{3}}{(A+2)(A-2)}}.} . The inverse function of the j-invariant can be expressed in terms of the hypergeometric function 2F1 (see also the article Picard–Fuchs
J-invariant
INVERSE FUNCTION
INVERSE FUNCTION
Girl/Female
Indian
Universe
Girl/Female
Muslim
Universe
Girl/Female
Muslim
Universe
Girl/Female
Hebrew
Incense.
Boy/Male
Tamil
Universe
Boy/Male
Tamil
Universe
Boy/Male
Hindu
Universe
Girl/Female
Indian
Universe
Girl/Female
Tamil
Universe
Surname or Lastname
Danish and Norwegian
Danish and Norwegian : patronymic from the personal name Ivar, from Old Norse Ãvarr, a compound of either Ãv ‘yew tree’, ‘bow’ or Ing (the name of a god) + ar ‘warrior’ or ‘spear’.North German (Frisian) : patronymic from a Germanic personal name composed of the elements Ä«wa ‘yew (tree)’ + hard ‘strong’, ‘firm’.English : variant spelling of Iverson.
Girl/Female
Australian, Greek
Kind; Innocent
Boy/Male
Hindu
Universe
Surname or Lastname
English
English : from Middle English, Old French convers ‘convert’ (Latin conversus, past participle of convertere ‘to turn’), hence a nickname for a Jew converted to Christianity, or more often an occupational name for someone converted to the religious way of life, a lay member of a convent.
Boy/Male
Muslim
Universe
Girl/Female
Muslim
Universe
Boy/Male
Tamil
Universe
Girl/Female
Greek
Kind or innocent.
Girl/Female
Hebrew
Incense.
Boy/Male
Indian
Universe
Boy/Male
Tamil
Universe
INVERSE FUNCTION
INVERSE FUNCTION
Female
Scandinavian
 Scandinavian pet form of Greek Hanna, ANNI means "favor; grace." Compare with another form of Anni.
Female
Hebrew
Variant spelling of Hebrew Bathsheba, BATSHEBA means "daughter of the oath."
Female
Irish
Pet form of Irish Gaelic Máire, MÃIRÃN means "obstinacy, rebelliousness" or "their rebellion."
Girl/Female
Hindu, Indian, Marathi, Sanskrit
A Lute; Instrument of Goddess Saraswati
Surname or Lastname
English
English : nickname for a bald man, from a diminutive of Anglo-Norman French cauf.
Boy/Male
Hindu, Indian, Punjabi, Sikh
Jewel
Surname or Lastname
English (mainly southern)
English (mainly southern) : from a Middle English personal name, a survival of Old English GÅdcild, composed of the elements gÅd ‘good’ + the late Old English name-forming element cild (see Child). This name may also have been used in the Middle English period as a nickname for a good person.English : nickname from godchild, i.e. someone who was the godchild of an important member of the community. Compare Godson, which was similarly confused with Goodson.English translation of German Gutkind (see Gutkin).
Boy/Male
French, German
Man; Strong and Masculine
Girl/Female
Indian
The earth
Boy/Male
Sikh
New Sun rays of motivation
INVERSE FUNCTION
INVERSE FUNCTION
INVERSE FUNCTION
INVERSE FUNCTION
INVERSE FUNCTION
a.
Opposite in nature and effect; -- said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity; as, multiplication is the inverse operation to division. The symbol of an inverse operation is the symbol of the direct operation with -1 as an index. Thus sin-1 x means the arc whose sine is x.
a.
The back side; as, the reverse of a drum or trench; the reverse of a medal or coin, that is, the side opposite to the obverse. See Obverse.
a.
Alt. of Renverse
adv.
In an inverse order or manner; by inversion; -- opposed to directly.
n.
To offer incense to. See Incense.
a.
Opposite in order, relation, or effect; reversed; inverted; reciprocal; -- opposed to direct.
v. t.
To reverse.
a.
Extreme in degree; excessive; immoderate; as: (a) Ardent; fervent; as, intense heat. (b) Keen; biting; as, intense cold. (c) Vehement; earnest; exceedingly strong; as, intense passion or hate. (d) Very severe; violent; as, intense pain or anguish. (e) Deep; strong; brilliant; as, intense color or light.
a.
Strained; tightly drawn; kept on the stretch; strict; very close or earnest; as, intense study or application; intense thought.
n.
An inverted arch.
v. t.
See Inhearse.
a.
Inverted; having a position or mode of attachment the reverse of that which is usual.
imp. & p. p.
of Invert
n.
That which is inverse.
a.
Subjected to the process of inversion; inverted; converted; as, invert sugar.
n.
To perfume with, or as with, incense.
a.
Reversed; as, a reverse shell.
a.
To turn upside down; to invert.
a.
Acting against, or in a contrary direction; opposed; contrary; opposite; conflicting; as, adverse winds; an adverse party; a spirit adverse to distinctions of caste.
a.
In hostile opposition to; unfavorable; unpropitious; contrary to one's wishes; unfortunate; calamitous; afflictive; hurtful; as, adverse fates, adverse circumstances, things adverse.