The function represents the correspondence between each input value and a unique output value. The standard symbol of the output value x corresponding to the input value in the function f is f(x). The set containing all the input values of a function is called the domain of this function, and the set containing all the output values is called the range. If you define the concept of mapping first, you can simply define a function as, and the mapping defined between non-empty number sets is called a function.

Classical definition of mathematical definition of catalogue: modern definition: mapping definition: introduction to computer definition: definition of concept mapping related to function, definition domain of set theory, injectivity of corresponding domain and value domain of geometric meaning function, development history of concepts of surjection and bijection function, boundedness of image property function, singularity of function, continuity of periodic function, concavity and convexity of real function or imaginary function, function concept in early 18th century, modern function concept in 19th century, Classification of inverse function, implicit function, multivariate function, quadratic function, transcendental function, power function and complex variable function according to unknowns, Function introduction C. Some functions in the language C. Generation conditions in the library function compound function definition language The definition domain periodically increases or decreases the image properties of specific functions commonly used in mathematics. Create function formulas in Word to expand the classical definition of mathematical definition: modern definition: definition by mapping: introduction to computer definition Concept mapping and function-related definition set theory definition domain, corresponding domain of geometric meaning function and injectivity of value domain, The development histWordy of the concepts of surjective and bijective functions, the boundedness of image property function, the singularity of function, the continuity of periodic function, the concavity and convexity of real function or imaginary function, the concept of function in the early 18th century, the concept of modern function in the 19th century, the inverse function of special function, implicit function, multivariate function, quadratic function, transcendental function, power function and complex variable function, and the classification of functions according to unknowns, and the function introduction of some function libraries, functions and compound functions in C.C. language.

Edit the classic definition of the mathematical definition in this paragraph: there are two variables X and Y in a certain change process. According to certain corresponding rules, for each given X value, there is a unique and definite Y value corresponding to it, so Y is a function of X, where X is the independent variable and Y is the dependent variable.

In addition, if there is a unique value of X for each given value of Y, then X is also a function of Y. ..

Modern definition: Generally speaking, given a set of non-null numbers A and B, according to a certain correspondence rule F, any element X in A has a unique Y in B, then this correspondence from set A to set B is called a function from set A to set B.

Note: x → y = f (x), x ∈ a. Set A is called the definition domain of the function, D, set {y ∣ y = f (x), x ∈ a} is called the range, C. Definition domain, range, and corresponding rules are called the three elements of the function. Generally written as y = f(x)x∈d, if the definition field is omitted, it refers to the set of all real numbers that are meaningful to the function.

Use the definition of mapping: Generally speaking, given non-null sets A and B, the mapping from set A to set B is called a function from set A to set B..

Vector function: a function whose independent variable is a vector is called vector function f(a 1.a2, a3 ... an) = y.

The important relationship between correspondence, mapping and function;

A function is a mapping on a set of numbers, and a mapping is a specific correspondence. That is, {function} is contained in {mapping} and {correspondence}.

These statements are used to complete some meaningful work in the process of editing this computer-defined function-usually processing text, controlling input or calculating numerical values. By introducing the function name and required parameters into the program code, the function can be executed (or called) in the program.

Similar to a procedure, but a function usually has a return value. They can all call themselves in their own structures, which is called recursion.

Most programming languages have function keywords (or reserved words) in the method of constructor.

Similar to mathematical functions, functions are often used in equations, such as y=f(x)(f is defined by users).

Editing this passage is a basic concept in mathematics and one of the most important concepts in algebra.

First of all, we should understand the corresponding relationship between functions and non-empty number sets. Then, understand that there is more than one functional relationship between A and B, and finally, focus on understanding the three elements of the function.

The corresponding rules of functions are usually expressed by analytical expressions, but a large number of functional relationships can not be expressed by analytical expressions, but only by images, tables and other forms.

Edit the concepts related to functions in this paragraph. In the process of change, the amount of change is called a variable, and some values do not change with the variable. We call it a constant.

Independent variable, function, is a variable related to other quantities, and any value in this quantity can find a corresponding fixed value in other quantities.

The dependent variable (function) changes with the change of the independent variable. When the independent variable takes a unique value, the dependent variable (function) has and only has a unique value corresponding to it.

Function value, where y is x, x determines a value, and y determines a value accordingly. When x takes a, y is determined as b, and b is called the function value of a. ..

The mapping definition assumes that A and B are two non-empty sets. If there is a unique element B corresponding to any element A in set A according to a certain correspondence F, then such correspondence (including sets A and B, and the correspondence F from set A to set B) is called the mapping from set A to set B, which is denoted as F: A → B, where b is called the image of A under the mapping F, and is denoted as b = f (a); A is called the original image of B with respect to the mapping f, and the set of images of all elements in set A is denoted as f(A).

Then there is: the mapping defined between non-empty number sets is called a function. The independent variable of a function is a special original image and the dependent variable is a special image.

Geometric meaning function is related to inequality and equation (elementary function). Let the function value be equal to zero, and from a geometric point of view, the corresponding independent variable value is the abscissa of the intersection of the image and the X axis; From the algebraic point of view, the corresponding independent variable is the solution of the equation. In addition, replacing "=" in the expression of a function (except a function without expression) with "",and then replacing "y" with other algebraic expressions, the function becomes an inequality, and the value range of the independent variable can be found.

Set theory of functions If the binary relation f: x× y from x to y has a unique y∈Y for each x∈X, so that < x, y & gt∈f, then call f a function from x to y and write it as: f: x → y.

When x = x = x1××× xn, f is called an n-ary function.

Its characteristics:

Pre-domain and domain overlap.

Unit price:

Edit the input value set x of the definition field, corresponding field and value field of this paragraph, which is called the definition field of f; The set y of possible output values is called the range of f. The range of a function refers to the set of actual output values obtained by mapping f to all elements in the definition field. Note that it is incorrect to call the corresponding domain, and the function's domain is a subset of the corresponding domain.

In computer science, the data types of parameters and return values determine the definition domain and corresponding domain of subroutines respectively. Therefore, the domain and the corresponding domain are mandatory constraints determined at the beginning of the function. On the other hand, the scope is related to the actual implementation.

Edit the injective function, injective function and injective function of this paragraph, and map different variables into different values. That is to say, if X and Y belong to the domain, f(x) is not equal to f(y) only when X is not equal to Y. ..

The range of injective surjective function is its corresponding range. That is, for any y in the mapping domain of mapping f, there is at least one x that satisfies f (x) = y.

A bijection function is both injective and injective. Also called one-to-one correspondence. Bijective function is often used to indicate that sets X and Y are equipotential, that is, they have the same cardinality. If two sets can establish a one-to-one correspondence, they are said to be equipotential.

Edit this image and the original image element x∈X. The image in F is f(x), and the formula they take is 0.

Subset a? The image of X in F is a subset of Y composed of images of its elements, that is, F (

Edit the function image in this paragraph The image of function f is a set of point pairs (x, f(x)) on the plane, where x takes all members in the domain. Function images are helpful to understand and prove some theorems.

If both x and y are continuous straight lines, the image of the function will be very intuitive. Note that the binary relation between two sets X and Y has two definitions: one is a triple (X, Y, G), where G is the graph of the relation; The second is to simply define the relationship with the graph. According to the second definition, the function f is equal to its image.

When k < 0, the straight line rises, passing through one or three quadrants or translating quadrants up and down; When k>0, the straight line is descending, passing through two or four quadrants, and translating the quadrants up or down.

Edit the boundedness of the attribute function in this paragraph. Let the domain of the function f(x) be D, and the number set X is contained in D. If the number K 1 exists, so that f(x)≤K 1 holds for any x∈X, then the function f(x) is said to have an upper bound on x, and K 1 is called the function F. If there is a number K2 that makes f(x)≥K2 hold for any x∈X, then the function f(x) has a lower bound on X, and K2 is called a lower bound on X. If there is a positive number M, then | f (x) |

The necessary and sufficient condition for the function f(x) to be bounded on X is that it has both upper and lower bounds on X.

The monotonicity of the function makes the domain of the function f(x) D, and the interval I is contained in D. If for any two points on the interval I, x 1 and x2, when X 1

The parity of a function Let f(x) be a real function, then f is a odd function. The following equation applies to all real numbers x:

F(x) = f(-x) or f( -x) =-f(x) Geometrically, a odd function is symmetrical with the origin, that is, its graph will not change after rotating 180 degrees around the origin.

Odd function's examples are X, sin(x), sinh(x) and erf(x).

Let f(x) be a real variable function, then f is an even function, if the following equation holds for all real numbers x:

F(x) = f(-x) Geometrically, an even function will be symmetrical about the Y axis, that is, its graph will not change after being mirrored on the Y axis.

Examples of even functions are |x|, x 2, cos(x) and cosh(sec)(x).

Even functions cannot be bijective mappings.

The periodic Dirichlet function of function Let the domain of function f(x) be D. If there is a positive number L, and (x ∈ l)∈D and f(x+l)=f(x) are constants, then f(x) is called a periodic function, and l is called f(x). Usually we say that the period of a periodic function refers to the minimum positive period. The domain D of a periodic function is an unbounded interval with at least one side. If d is bounded, the correction function is not periodic.

Not every periodic function has a minimum positive period, such as Dirichlet function.

Continuity of Function In mathematics, continuity is an attribute of function. Intuitively, a continuous function is a function in which the change of the input value is small enough and the change of the output is small enough. If a small change in the input value will cause a sudden jump, or even the output value is uncertain, the function is called a discontinuous function (or discontinuous function).

Let f be a function projected from a subset of a set of real numbers. F is continuous at point C if and only if the following two conditions are met:

F is defined at point C, and C is a convergence point in in. No matter how the independent variable X approaches C in In, the limit of f(x) exists and is equal to f(c). We say that a function is continuous everywhere or everywhere, or if it is continuous at any point in its definition domain, it is simply continuous. More generally, we say that a function is continuous on a subset of its domain, when it is continuous at every point on this subset.

Without the concept of limit, the continuity of real function can also be defined by the following so-called method.

Let's consider the function. Suppose c is an element in the domain of f. The function f is continuous at point C if and only if the following conditions are true:

For any positive real number, there is a positive real number δ >; 0, so for any domain, as long as x satisfies c-δ.

Let f(x) be continuous on I, if there are two points x 1≠x2 on I, there will always be F((x 1+x2)/2) ≤ (f (x1)+f (x2))/2, (. (f(x 1)+f(x2))/2) and then say that f(x) is a (strictly) convex function on the interval I; If f ((x1+x2)/2) ≥ (f(x1)+f (x2))/2, (f ((x1+x2)/2) > (f (x1x2))

Real function or virtual function Real function refers to a function whose domain and value domain are real numbers. One of the characteristics of real functions is that they can draw pictures on coordinates.

Virtual function is an important concept in object-oriented programming. When inheriting from the parent class, the virtual function and the inherited function have the same signature. However, in the process of running, the running system will automatically choose the appropriate specific implementation to run according to the type of object. Virtual function is the basic means to realize polymorphism by object-oriented programming.

Edit the development history of functional concepts in this paragraph. 1Galileo (Italy, 1564- 1642) in the 7th century almost all contained the concept of the relationship between functions or variables, and expressed the relationship between functions in the language of words and proportions. Descartes (France, 1596- 1650) noticed the dependence of one variable on another in Analytic Geometry around 1637, but he didn't realize the need to refine the concept of function at that time, so Newton and Leibniz established it in the late17th century.

1673, Leibniz first used "function" to express "power". Later, he used this word to represent the geometric quantities of each point on the curve, such as abscissa, ordinate, tangent length and so on. At the same time, Newton used "flow" to express the relationship between variables in the discussion of calculus.

/kloc-in the 8th century, johann bernoulli (Swiss, 1667- 1748) defined the concept of function on the basis of the concept of Leibniz function: "a quantity consisting of any variable and any form of constant." He means that any formula composed of variable X and constant is called a function of X, and he emphasizes that functions should be expressed by formulas. 1748, Bernoulli's student Euler said in his book Introduction to Infinite Analysis: "The function of a variable is an analytical expression composed of some numbers or constants of a variable and any way.

1755, Euler (L. Euler, Switzerland, 1707- 1783) defines a function as "if some variables depend on other variables in some way, that is, when the latter variable changes, the former variable also changes, and we call the former variable a function of the latter variable."

Euler (L. Euler, Switzerland, 1707- 1783) gave a definition: "The function of a variable is an analytical expression composed of this variable and some numbers or constants in any way." He called the function definition given by johann bernoulli's analytic function, and further divided it into algebraic function and transcendental function, which was also considered as "arbitrary function". It is not difficult to see that Euler's definition of function is more universal and extensive than johann bernoulli's.

In the19th century, the concept of function was 182 1 year. Cauchy (France, 1789- 1857) gave a definition from the definition of variables: "Some variables have certain relationships. When the value of one variable is given, the values of other variables can be determined accordingly, and the initial variable will be adopted. The word independent variable appeared for the first time in Cauchy's definition, and pointed out that functions don't need analytic expressions. However, he still believes that functional relationships can be expressed by multiple analytical expressions, which is a great limitation.

1822, Fourier (France,1768-1830) found that some functions have also been expressed by curves, or they can be expressed by one formula, or they can be expressed by multiple formulas, thus ending the debate on whether the concept of functions is expressed by only one formula and pushing the understanding of functions to a new level.

In 1837, Dirichlet (Germany, 1805- 1859) broke through this limitation and thought that it was irrelevant how to establish the relationship between x and y. He broadened the concept of function and pointed out: "For every definite value of X in a certain interval, Y has a definite value, so this definition of Y avoids the description of dependency in the definition of function and is accepted by all mathematicians in a clear way. This is what people often call the classic function definition.

After the set theory founded by Cantor (German, 1845- 19 18) occupied an important position in mathematics, veblen (American, veblen, 1880- 1960) used "set" and ".

Modern function concept1914 F. Hausdorf defined the function with the fuzzy concept of "ordered couple" in the Outline of Set Theory, which avoided the two fuzzy concepts of "variable" and "correspondence". In 192 1, Kuratowski defined "ordered pair" with the concept of set, which made Hausdorff's definition very strict.

In 1930, the new modern function is defined as "If there is always an element Y determined by set N corresponding to any element X of set M, then a function is defined on set M, and it is denoted as y=f(x). Element x is called an independent variable and element y is called a dependent variable. "

In general, let the range of the function y=f(x)(x∈A) be C. According to the relationship between x and y in this function, express x with y and get x= f(y). If for any value of y in C, X has a unique value and it is in A, the definition domain and value domain of the inverse function y = f- 1 (x) are the definition domain and value domain of the function y=f(x) respectively.

Note: (1) In the function x = f- 1 (y), y is the independent variable and x is the function, but traditionally we usually use x as the independent variable and y as the function, so we often switch the letters x and y in the function x = f- 1 (y) and rewrite it as y =.

Inverse function is also a function, because it conforms to the definition of function. It can be seen from the definition of inverse function that any function y=f(x) does not necessarily have an inverse function. If the function y=f(x) has an inverse function y = f- 1 (x), then the inverse function of the function y = f- 1 (x) is y=f(x). .

(3) From the definition of mapping, we can see that the function y=f(x) is the mapping from the domain A to the value domain C, and its inverse function y = f- 1 (x) is the mapping from the set C to the set A, so the domain of the function y=f(x) is just its inverse function y = f-/kloc-0. The range of the function y=f(x) is exactly the domain of its inverse function y = f- 1 (x) (as shown in the following table):

Function y=f(x) Inverse function y = f- 1 (x)

Definition domain A C

Range C A

(4) The above definition can be described by the concept of "inverse" mapping:

If it is determined that the mapping f of the function y=f(x) is a "one-to-one mapping" from the definition domain to the value domain, then the function x = f- 1 determined by the "inverse" mapping f is called the inverse function of the function y=f(x). .

The first two examples: s=vt is recorded as f(t)=vt, then its inverse function can be written as f- 1 (t) = t/v, and y=2x+6 is recorded as f(x)=2x+6, then its inverse function is: f- 1 (.

Inverse functions sometimes need to be discussed in categories, such as: f(x)=X+ 1/X, and x needs to be discussed in categories: when x is greater than 0, x is less than 0, which needs attention. The inverse function of a general fractional function is expressed as y=ax+b/cx+d(a/c is not equal to b/d)-y = b-dx/CX+a.

Application of inverse function;

When it is difficult to find the range of the function directly, the range of the original function can be determined by finding the definition range of the original function. The steps to find the inverse function are as follows:

1. Find the range of the original function first, because the range of the original function is the definition range of the inverse function.

We know that the three elements of a function are domain, range and corresponding rules, so finding the domain of the inverse function first is the first step to find the inverse function.

2. inverse solution x, that is, x is represented by y.

To rewrite the exchange position is to change x into y and y into X.

4. Write the inverse function and its domain.

As far as the relationship is concerned, it is generally two-way, and so is the function. Let y=f(x) be a known function. If every y has a unique x∈X, so that f(x)=y, it is the process of finding x from y, that is, x becomes a function of y, and it is recorded as x = f-65438. Then f-1 is the inverse function of f, and x is traditionally used to represent the independent variable, so this function is still recorded as y=f-1(x). For example, y=sinx and y=arcsinx are reciprocal functions. In the same coordinate system, the graphs of y=f(x) and y=f-1(x) are symmetrical about the straight line y = x.

If the implicit function can be determined by the equation f(x, y)=0, Y is the function y=f(x, that is, F(x, f(x))≡0, then Y is said to be the implicit function of X.

Note: The equation F(x, y )= 0 here is not a function.

Thinking: Is implicit function a function?

No, because it is not satisfied with "one-to-one" and "many-to-one" in the process of change.

Set point of multivariate function (x 1, x2, …, xn) ∈G? Rn，U？ R 1, if there is a unique u∈U corresponding to each point (x 1, x2, ..., xn)∈G f: g→ u, u=f(x 1, x2, ..., xn.

Basic elementary functions and their images such as power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function are called basic elementary functions.

① Power function: y = x μ (μ ≠ 0, μ is any real number) Definition domain: μ is a positive integer: (-∞, +∞), μ is a negative integer: (-∞, 0)∞(0, +∞); μ=α(a is an integer), (-∞, +∞) when α is odd and (0, +∞) when α is even; μ=p/q, p, q is coprime, and is a composite function of. Sketches are shown in Figures 2 and 3.

② exponential function: y = a x (a > 0, a≠ 1), the domain is (-∞, +∞), the range is (0, +∞), and a> 1 is a strictly monotonic increasing function (i.e. when x2 >；; When x 1,) 0③ logarithmic function: y = logax(a & gt；; 0), called a as the base, the domain is (0, +∞), and the range is (-∞, +∞). A> 1 is strictly monotonically increasing, while 0

Logarithms with the base of 10 are called ordinary logarithms and abbreviated as lgx. Logarithms based on e are widely used in science and technology, that is,

④ Trigonometric function: See Table 2.

Sine function and cosine function are shown in figs. 6 and 7.

⑤ Inverse trigonometric function: See Table 3. Hyperbolic sine and cosine are shown in figure 8.

⑥ Hyperbolic function: hyperbolic sine (ex-e-x), hyperbolic cosine? (ex+e-x), hyperbolic tangent (ex-e-x)/(ex+e-x), hyperbolic cotangent (ex+e-x)/(ex-e-x).

Edit this paragraph and classify the constant function x according to the number of unknowns. When you take any number in the definition field, there is y=C (C is a constant), then the function y=C is called a constant function, which is like a straight line or a part of a straight line parallel to the X axis. First, the definition and definition formula: the independent variable X and the dependent variable Y have the following relationship: y=kx+b(k, B is a constant, k≠0), then Y is said to be a first-order function of X, especially when b=0, that is, y=kx, and Y is a proportional function of X.

Two. Properties of linear function: the change value of y is directly proportional to the corresponding change value of x, and the ratio is k, that is, y/x = kⅲ. Images and properties of linear functions;