Infix notation
53-633: HP-17B is an algebraic entry financial and business calculator manufactured by Hewlett-Packard , introduced on 4 January 1988 along with the HP-19B , HP-27S and the HP-28S . It was a simplified business model, like the 19B. There were two versions, the US one working in English only, and the international one with a choice of six languages (English, German , Spanish , French , Italian , and Portuguese ). HP-17B code name
106-511: A dual pair to show the underlying duality . This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation , the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application . In category theory and homological algebra , networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize
159-406: A function from a set X to a set Y assigns to each element of X exactly one element of Y . The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically , the concept
212-432: A map or a mapping , but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds ). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H ). Some authors reserve
265-405: A roman type is customarily used instead, such as " sin " for the sine function , in contrast to italic font for single-letter symbols. The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let f ( x ) {\displaystyle f(x)} be a function". This is an abuse of notation that is useful for
318-429: A function defined by an integral with variable upper bound: x ↦ ∫ a x f ( u ) d u {\textstyle x\mapsto \int _{a}^{x}f(u)\,du} . There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis , linear forms and the vectors they act upon are denoted using
371-401: A function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details. A function f from a set X to a set Y is an assignment of one element of Y to each element of X . The set X is called the domain of the function and the set Y is called the codomain of
424-523: A function is commonly written as f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as f ( w , x , y ) {\displaystyle f(w,x,y)} , f ( w , x , y , z ) {\displaystyle f(w,x,y,z)} . A function may also be called
477-513: A function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is a real function , the determination of the domain of the function x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} requires knowing
530-405: A function is then called a partial function . The range or image of a function is the set of the images of all elements in the domain. A function f on a set S means a function from the domain S , without specifying a codomain. However, some authors use it as shorthand for saying that the function is f : S → S . The above definition of a function is essentially that of
583-461: A function notation would be S (1, 3) in which the function S denotes addition ("sum"): S(1, 3) = 1 + 3 = 4 . In infix notation, unlike in prefix or postfix notations, parentheses surrounding groups of operands and operators are necessary to indicate the intended order in which operations are to be performed. In the absence of parentheses, certain precedence rules determine the order of operations . Function (mathematics) In mathematics ,
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#1733104268295636-552: A letter such as f , g or h . The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x ) is denoted by f ( x ) ; for example, the value of f at x = 4 is denoted by f (4) . Commonly, a specific function is defined by means of an expression depending on x , such as f ( x ) = x 2 + 1 ; {\displaystyle f(x)=x^{2}+1;} in this case, some computation, called function evaluation , may be needed for deducing
689-417: A multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain. where the domain U has the form If all the X i {\displaystyle X_{i}} are equal to the set R {\displaystyle \mathbb {R} } of the real numbers or to the set C {\displaystyle \mathbb {C} } of
742-450: A simpler formulation. Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced " maps to ". For example, x ↦ x + 1 {\displaystyle x\mapsto x+1} is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of R {\displaystyle \mathbb {R} }
795-411: Is a stub . You can help Misplaced Pages by expanding it . Infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands —"infixed operators"—such as the plus sign in 2 + 2 . Binary relations are often denoted by an infix symbol such as set membership a ∈ A when
848-423: Is a function in two variables, and we want to refer to a partially applied function X → Y {\displaystyle X\to Y} produced by fixing the second argument to the value t 0 without introducing a new function name. The map in question could be denoted x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} using
901-400: Is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. Formally, a function of n variables is a function whose domain is a set of n -tuples. For example, multiplication of integers is a function of two variables, or bivariate function , whose domain is
954-479: Is called the Cartesian product of X and Y and denoted X × Y . {\displaystyle X\times Y.} Thus, the above definition may be formalized as follows. A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions: This definition may be rewritten more formally, without referring explicitly to
1007-438: Is implied. The domain and codomain can also be explicitly stated, for example: This defines a function sqr from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f : X × X → Y ; ( x , t ) ↦ f ( x , t ) {\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}
1060-432: Is in Y , or it is undefined. The set of the elements of X such that f ( x ) {\displaystyle f(x)} is defined and belongs to Y is called the domain of definition of the function. A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X , one often says that
1113-419: Is more difficult to parse by computers than prefix notation (e.g. + 2 2) or postfix notation (e.g. 2 2 + ). However many programming languages use it due to its familiarity. It is more used in arithmetic, e.g. 5 × 6. Infix notation may also be distinguished from function notation, where the name of a function suggests a particular operation, and its arguments are the operands. An example of such
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#17331042682951166-451: Is often the letter f . Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in The argument between the parentheses may be a variable , often x , that represents an arbitrary element of the domain of the function, a specific element of the domain ( 3 in
1219-440: Is the value of the function at x , or the image of x under the function. A function f , its domain X , and its codomain Y are often specified by the notation f : X → Y . {\displaystyle f:X\to Y.} One may write x ↦ y {\displaystyle x\mapsto y} instead of y = f ( x ) {\displaystyle y=f(x)} , where
1272-469: Is typically the case for functions whose domain is the set of the natural numbers . Such a function is called a sequence , and, in this case the element f n {\displaystyle f_{n}} is called the n th element of the sequence. The index notation can also be used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during
1325-442: The graph of the function , a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. Functions are widely used in science , engineering , and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics. The concept of
1378-687: The Riemann hypothesis . In computability theory , a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem ). A multivariate function , multivariable function , or function of several variables
1431-413: The complex numbers , one talks respectively of a function of several real variables or of a function of several complex variables . There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below. The functional notation requires that a name is given to the function, which, in the case of a unspecified function
1484-420: The zeros of f. This is one of the reasons for which, in mathematical analysis , "a function from X to Y " may refer to a function having a proper subset of X as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers , typically a subset that contains a non-empty open interval . Such
1537-550: The "total" condition removed. That is, a partial function from X to Y is a binary relation R between X and Y such that, for every x ∈ X , {\displaystyle x\in X,} there is at most one y in Y such that ( x , y ) ∈ R . {\displaystyle (x,y)\in R.} Using functional notation, this means that, given x ∈ X , {\displaystyle x\in X,} either f ( x ) {\displaystyle f(x)}
1590-452: The above example), or an expression that can be evaluated to an element of the domain ( x 2 + 1 {\displaystyle x^{2}+1} in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let f ( x ) = sin ( x 2 + 1 ) {\displaystyle f(x)=\sin(x^{2}+1)} ". When
1643-432: The arrow notation for functions described above. In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function f can be defined as mapping any pair of real numbers ( x , y ) {\displaystyle (x,y)} to the sum of their squares, x 2 + y 2 {\displaystyle x^{2}+y^{2}} . Such
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1696-590: The arrow notation. The expression x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f ( x 0 , t 0 ) refers to the value of the function f at the point ( x 0 , t 0 ) . Index notation may be used instead of functional notation. That is, instead of writing f ( x ) , one writes f x . {\displaystyle f_{x}.} This
1749-434: The concept of a relation, but using more notation (including set-builder notation ): A function is formed by three sets, the domain X , {\displaystyle X,} the codomain Y , {\displaystyle Y,} and the graph R {\displaystyle R} that satisfy the three following conditions. Partial functions are defined similarly to ordinary functions, with
1802-476: The domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets X and Y is a subset of the set of all ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x\in X} and y ∈ Y . {\displaystyle y\in Y.} The set of all these pairs
1855-410: The domain of definition of a complex function is illustrated by the multiplicative inverse of the Riemann zeta function : the determination of the domain of definition of the function z ↦ 1 / ζ ( z ) {\displaystyle z\mapsto 1/\zeta (z)} is more or less equivalent to the proof or disproof of one of the major open problems in mathematics,
1908-448: The domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse. Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set C {\displaystyle \mathbb {C} } of the complex numbers . The difficulty of determining
1961-408: The founders of calculus , Leibniz , Newton and Euler . However, it cannot be formalized , since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory . This set-theoretic definition is based on the fact that a function establishes a relation between the elements of
2014-474: The function f (⋅) from its value f ( x ) at x . For example, a ( ⋅ ) 2 {\displaystyle a(\cdot )^{2}} may stand for the function x ↦ a x 2 {\displaystyle x\mapsto ax^{2}} , and ∫ a ( ⋅ ) f ( u ) d u {\textstyle \int _{a}^{\,(\cdot )}f(u)\,du} may stand for
2067-407: The function. If the element y in Y is assigned to x in X by the function f , one says that f maps x to y , and this is commonly written y = f ( x ) . {\displaystyle y=f(x).} In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable , and the corresponding element of Y
2120-436: The notation x ↦ f ( x ) , {\displaystyle x\mapsto f(x),} the symbol x does not represent any value; it is simply a placeholder , meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an interpunct " ⋅ ". This may be useful for distinguishing
2173-404: The partial function is a total function . In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain. In calculus , a real-valued function of a real variable or real function is a partial function from
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2226-408: The set R {\displaystyle \mathbb {R} } of the real numbers to itself. Given a real function f : x ↦ f ( x ) {\displaystyle f:x\mapsto f(x)} its multiplicative inverse x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} is also a real function. The determination of
2279-538: The set A has a for an element. In geometry , perpendicular lines a and b are denoted a ⊥ b , {\displaystyle a\perp b\ ,} and in projective geometry two points b and c are in perspective when b ⩞ c {\displaystyle b\ \doublebarwedge \ c} while they are connected by a projectivity when b ⊼ c . {\displaystyle b\ \barwedge \ c.} Infix notation
2332-795: The set of all n -tuples ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} such that x 1 ∈ X 1 , … , x n ∈ X n {\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}} is called the Cartesian product of X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},} and denoted X 1 × ⋯ × X n . {\displaystyle X_{1}\times \cdots \times X_{n}.} Therefore,
2385-865: The set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation . Commonly, an n -tuple is denoted enclosed between parentheses, such as in ( 1 , 2 , … , n ) . {\displaystyle (1,2,\ldots ,n).} When using functional notation , one usually omits the parentheses surrounding tuples, writing f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} instead of f ( ( x 1 , … , x n ) ) . {\displaystyle f((x_{1},\ldots ,x_{n})).} Given n sets X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},}
2438-594: The study of a problem. For example, the map x ↦ f ( x , t ) {\displaystyle x\mapsto f(x,t)} (see above) would be denoted f t {\displaystyle f_{t}} using index notation, if we define the collection of maps f t {\displaystyle f_{t}} by the formula f t ( x ) = f ( x , t ) {\displaystyle f_{t}(x)=f(x,t)} for all x , t ∈ X {\displaystyle x,t\in X} . In
2491-443: The symbol ↦ {\displaystyle \mapsto } (read ' maps to ') is used to specify where a particular element x in the domain is mapped to by f . This allows the definition of a function without naming. For example, the square function is the function x ↦ x 2 . {\displaystyle x\mapsto x^{2}.} The domain and codomain are not always explicitly given when
2544-447: The symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin( x ) . Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case,
2597-432: The value of the function at a particular value; for example, if f ( x ) = x 2 + 1 , {\displaystyle f(x)=x^{2}+1,} then f ( 4 ) = 4 2 + 1 = 17. {\displaystyle f(4)=4^{2}+1=17.} Given its domain and its codomain, a function is uniquely represented by the set of all pairs ( x , f ( x )) , called
2650-504: The word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as Serge Lang , use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions. In the theory of dynamical systems , a map denotes an evolution function used to create discrete dynamical systems . See also Poincaré map . Whichever definition of map
2703-519: Was Trader and it belonged to the Pioneer series of Hewlett-Packard calculators. It had a 131×16 LCD dot matrix , 22×2 characters, menu-driven display, used a Saturn processor and had a memory of 8000 bytes , of which 6750 bytes were available to the user for variable and equation storage. The HP-17B had a clock with alarm that allowed for basic agenda capabilities, as well an infrared port for printing to some Hewlett-Packard infrared printers . The 17B
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#17331042682952756-415: Was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory , and this greatly increased the possible applications of the concept. A function is often denoted by
2809-501: Was replaced by the HP 17BII (F1638A) (code name Trader II ) in January 1990, which added RPN entry. The 17BII was replaced by the HP 17bII+ in 2003. Two significantly different case variants of the 17bII+ exist. The newer 17bII+ (F2234A), introduced in 2007, with Sunplus Technology SPLB31A CPU was developed and is manufactured by Kinpo Electronics . This technology-related article
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