In the field of mathematics, a function is a relationship that makes each element in one set correspond to the only element in another (possibly the same) set.

This is only the case of unary function f (x) = y, please give a general definition according to the original English text, thank you.

-A variable that is related to another variable, and each value of a variable has another definite value.

Independent variable, a variable whose function is related to another quantity. Any value in this quantity can find a corresponding fixed value in another quantity.

A correspondence rule between two sets such that a unique element in the second set is assigned to each element in the first set.

The law of one-to-one correspondence between two groups of elements of a function, each element in the first group has only a unique corresponding quantity in the second group.

The concept of function is the most basic for every branch of mathematics and quantity.

function

Correspondence in mathematics is the correspondence from real number set A to real number set B. Simply put, A varies with B, and A is a function of B. To be precise, let X be an empty set, Y a real set, and F a rule. If every X in X has a Y corresponding to it according to the rule F, it is said that F is a function on X, denoted as Y = f(x), X is the domain of the function f(x), Y is its range, X is the independent variable and Y is the cause.

Example1:y = sinxx = [0,2π], y = [- 1, 1], and a functional relationship is given. Of course, changing y to y 1 = (a, b), where a < b is an arbitrary real number, is still a functional relationship.

The corresponding relationship between its depth y and the distance x from the shore point o to the measurement point is curved, which represents a function with the domain [0, b]. The above three examples show three representations of functions: formula method, table method and image method.

compound function

There are three variables, y is a function of u, y = ψ (u), u is a function of x, and u = f (x). These three variables often form a chain: y forms a function of x through the intermediate variable u:

X→u→y, which depends on the domain: let the domain of ψ be U. f and the value domain be u, when U*? U, f, ψ is said to form a composite function, such as y = lgsinx, x∈(0, π). At this time, sinx > 0, lgsinx is meaningful. But if x ∈ (-π, 0) is specified, then SINX < 0, lgsinx is meaningless and cannot be a composite function.

inverse function

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. F-1 is the inverse function of F. Traditionally, X is 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.

implicit function

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

Multivariate function

Setpoint (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 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 = ax (a > 0, a≠ 1), defined as (-∞, +∞), with a range of (0, +∞), which is strictly monotonic when a > 0 (i.e. x2 > x 1, 0 < a). For any A, the image passes through the point (0, 1). Note that the graphs of y = ax and y = () x are symmetrical about y axis. As shown in figure 4.

③ logarithmic function: y = logax (a > 0), where a is the base, the domain is (0, +∞), and the range is (-∞, +∞). A > 1 strictly monotonically increases, and 0 < A < 1 strictly monotonically decreases. No matter what the value of a is, the graph of logarithmic function passes through the point (1, 0), and both logarithmic function and exponential function are reciprocal functions. As shown in fig. 5.

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

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] supplement

In the field of mathematics, a function is a relationship, which makes each element in one set correspond to the only element in another (possibly the same) set (this is only the case of unary function f (x) = y, please give a general definition according to the original English text, thank you). The concept of function is the most basic for every branch of mathematics and quantity.

The terms function, mapping, correspondence and transformation usually have the same meaning.

quadratic function

I. Definition and definition of expressions

Generally speaking, there is the following relationship between independent variable x and dependent variable y:

y=ax？ +bx+c(a, b, c are constants, a≠0)

Y is called the quadratic function of X.

The right side of a quadratic function expression is usually a quadratic trinomial.

Two. Three Expressions of Quadratic Function

General formula: y=ax? +bx+c(a, b, c are constants, a≠0)

Vertex: y=a(x-h)? +k[ vertex P(h, k) of parabola]

Intersection point: y = a(X-X 1)(X-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]

Note: Among these three forms of mutual transformation, there are the following relations:

h=-b/2a k=(4ac-b？ )/4a x 1，x2=(-b √b？ -4ac)/2a

Three. Image of quadratic function

Do quadratic function y=x in plane rectangular coordinate system? Images,

It can be seen that the image of quadratic function is a parabola.

Four. Properties of parabola

1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line

x = -b/2a .

The only intersection of the symmetry axis and the parabola is the vertex p of the parabola.

Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).

2. The parabola has a vertex p, and the coordinates are

P [ -b/2a，(4ac-b？ )/4a ].

-b/2a=0, p is on the y axis; When δδ= b? When -4ac=0, p is on the x axis.

3. Quadratic coefficient A determines the opening direction and size of parabola.

When a > 0, the parabola opens upward; When a < 0, the parabola opens downward.

The larger the |a|, the smaller the opening of the parabola.

4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.

When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis;

When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.

5. The constant term c determines the intersection of parabola and Y axis.

The parabola intersects the Y axis at (0, c)

6. Number of intersections between parabola and X axis

δ= b？ When -4ac > 0, the parabola has two intersections with the x-axis.

δ= b？ When -4ac=0, the parabola has 1 intersections with the X axis.

δ= b？ When -4ac < 0, the parabola has no intersection with the x axis.

Verb (abbreviation of verb) quadratic function and unary quadratic equation

Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c，

When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).

Is that an axe? +bx+c=0

At this point, whether the function image intersects with the X axis means whether the equation has real roots.

The abscissa of the intersection of the function and the x axis is the root of the equation.

linear function

I. Definitions and definitions:

Independent variable x and dependent variable y have the following relationship:

Y=kx+b(k, b is a constant, k≠0)

It is said that y is a linear function of x.

In particular, when b=0, y is a proportional function of x.

Two. Properties of linear functions:

The change value of y is directly proportional to the corresponding change value of x, and the ratio is K.

That is △ y/△ x = K.

Three. Images and properties of linear functions;

1. exercises and graphics: through the following three steps (1) list; (2) tracking points; (3) Connecting lines can make images of linear functions-straight lines. So the image of a function only needs to know two points and connect them into a straight line.

2. Property: any point P(x, y) on the linear function satisfies the equation: y = kx+b.

3. Quadrant where k, b and function images are located.

When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;

When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.

When b > 0, the straight line must pass through the first and second quadrants; When b < 0, the straight line must pass through three or four quadrants.

Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.

At this time, when k > 0, the straight line only passes through one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.

Four. Determine the expression of linear function:

Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..

(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.

(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, two equations can be listed:

Y 1 = KX 1+B 1，Y2 = KX2+B2。

(3) Solve this binary linear equation and get the values of K and B. ..

(4) Finally, the expression of the linear function is obtained.

The application of verb (verb's abbreviation) linear function in life

1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.

2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t .. Set the original water quantity in the pool. G = S- feet.

inverse proportion function

A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.

The range of the independent variable x is all real numbers that are not equal to 0.

The image of the inverse proportional function is a hyperbola.

As shown in the figure, the function images when k is positive and negative (2 and -2) are given above.

trigonometric function

Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.

Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.

Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.

It has six basic functions:

Function name sine cosine tangent cotangent secant cotangent

Symbol sin cos tan cot sec csc

Sine function sin(A)=a/h

Cosine function cos(A)=b/h

Tangent function tan(A)=a/b

Cotangent function cot(A)=b/a

In a certain change process, the two variables X and Y, for each value of X within a certain range, Y has a certain value corresponding to it, and Y is a function of X. This relationship is generally expressed by y=f(x).

The development history of function concept

The Early Concept of 1. Function —— Function under the Concept of Geometry

/kloc-Galileo in the 0/7th century (Italy, 1564- 1642), in his book "Two New Sciences", 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 around his analytic geometry 1673. However, because he didn't realize that the concept of function needed to be refined at that time, no one had defined the function until Newton and Leibniz established calculus 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.

2.18th century function concept-function under algebraic concept.

John? Bernoulli John (Rui, 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.

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 put John? The definition of function given by Bernoulli is called analytic function, which is further divided into algebraic function and transcendental function, and "arbitrary function" is also considered. It is not difficult to see that Euler's definition of function is better than John's? Bernoulli's definition is more general and has a wider meaning.

3./kloc-the concept of function in the 0/9th century-the function under correspondence.

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, so the initial variable is called independent variable. At the same time, it is pointed out that analytic expressions are not necessary for functions. However, he still believes that functional relationships can be expressed by multiple analytical expressions, which is a great limitation.

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 one or more definite values. This definition avoids the description of dependence in function definition 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) played an important role in mathematics, veblen (American, veblen, 1880- 1960) used "set" and ".

4. Modern function concept-function under set theory

F. Hausdorff defined the function in 19 14 with the fuzzy concept of "ordered couple" in the outline of set theory, avoiding 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. "

The terms function, mapping, correspondence and transformation usually have the same meaning.

But the function only represents the correspondence between numbers, and the mapping can also represent the correspondence between points and between graphs. It can be said that the mapping contains functions.

Positive proportional function:

The image of the proportional function y=kx(k is constant, k≠0) is a straight line passing through the origin. When x >; 0, the image passes through three or one quadrant and rises from left to right, that is, y increases with the increase of x; When k < 0, the image passes through two or four quadrants and decreases from left to right, that is, y decreases with the increase of x.

It is precisely because the image of the proportional function y=kx(k is constant, k≠0) is a straight line that we can call it a straight line y=kx.

(another: the origin of the Chinese name "function"

In the book Algebra translated by China mathematician Li (181-1882), the word "function" was translated into Chinese for the first time, and this translation is still in use today. As for why the concept is translated in this way, the book explains that "whoever believes in this variable is a function of that variable"; "Faith" here means tolerance. )

An in-depth study of a function

Xu ruohan

When learning a function, according to the requirements of middle school, we should further study its practical application and how to change the position of the image.

1. Application of piecewise function in practical problems

(Example 1) (Wuhan, 2005) Xiaoming rides a bike from home to school in the morning, going uphill first and then downhill. The itinerary is as shown. How long will it take Xiao Ming to ride home from school if the speed of going up and down the hill remains the same when he returns?

Analysis: the speed of uphill and downhill is different, so the problem should be studied in two sections.

According to the information provided by the function image, we can know that when Xiaoming goes to school from home, the uphill distance is 3600 meters and the downhill distance is 9600-3600 = 6000 meters.

∴ uphill speed is 3600÷ 18=200 (m/min).

Downhill speed is 6000 ÷ (30- 18) = 500 (m/min).

When Xiao Ming came home, the uphill journey was 6000 meters, and the downhill journey was 3600 meters. It took 6000÷200+3600÷500 = 37.2 minutes.

Second, the application in physics.

[Example 2] (Huanggang City, 2004) When a class of students explored the relationship between spring length and external force, the corresponding data recorded in the experiment were as follows:

Find the resolution function of y about x and the range of independent variables.

Analysis: According to the knowledge of physics, the spring is deformed (elongated) under the action of external force (gravity of hanging heavy objects), and the relationship between external force and pointer position can be expressed by a linear function; But the external force on each spring is limited, so we must find the range of independent variables.

According to the known data, it is found that during the spring stretching process,

Let y=7.5 and get x=275.

The function of ∴ is

Note that the dividing point between two paragraphs is x=275, not x=300.

Third, the application of linear translation

Example 3 (Heilongjiang Province in 2005) In the rectangular coordinate system, points A (-9,0), P (0 0,3) and C (0, 12) are known. Q: Is there a point Q on the X-axis, so that a quadrilateral with points A, C, P and Q as its vertices is a trapezoid? If it exists, find the analytical formula of straight line PQ; If it does not exist, please explain why.

Analysis: Which two sides are parallel in the studied trapezoid? There are two possibilities: if, that is, the straight line CA is translated, the analytical formula of the straight line CA can be easily obtained through point P as follows.

The analytical formula of the straight line obtained after translation is

if

Translation line PA: through point C.

Get a straight line:

The line intersects the X axis at the point (-36,0).

The analytical formula of straight line is

How to understand the concept of function

Cao Yang

Function is an extremely important basic concept in mathematics. In middle school mathematics, functions and their related contents are very rich and occupy a great weight. Mastering the concept of function is very useful for future study. Looking back on the development history of the concept of function, Leibniz first adopted "function" as a mathematical term. He first put forward the concept of function in his paper 1692, but its meaning is quite different from the current understanding of function. In modern junior high school mathematics curriculum, the definition of function is "variable theory". Namely:

In a certain change process, there are two variables X and Y. If there is a unique definite value corresponding to each definite value of X within a certain range, then Y is called a function of X, X is called an independent variable, and Y is called a dependent variable.

It clearly points out that the independent variable X can take any value within a given range, and the dependent variable Y takes a unique and certain value every time according to certain laws. But junior high school does not require mastering the range of independent variables (just look at several functions to be learned in junior high school, and you will know that this definition is completely sufficient and easy to understand for junior high school students).

The concept of function is very abstract, which is difficult for students to understand. To understand the concept of function, we must clarify two points: First, we must clarify the relationship between independent variables and dependent variables. In a certain change process, there are two variables X and Y. If Y changes with X, then X is called independent variable and Y is called dependent variable. If x changes with y, then y is called independent variable and x is called dependent variable. Second, the core of function definition is "one-to-one correspondence", that is, given the value of an independent variable X, there is a uniquely determined value of a dependent variable Y corresponding to it. Such correspondence can be "one independent variable corresponds to one dependent variable" (abbreviated as "one-to-one") or "several independent variables correspond to one dependent variable" (abbreviated as "many-to-one"), but it cannot be "one independent variable corresponds to multiple dependent variables".

One-to-one, many-to-one and one-to-many.

It's a function, a function, not a function

Figure 1

Here are four examples to help you understand the concept of function:

Example 1 The length of the spring is 10 cm. When the spring is pulled by F(F is within a certain range), the length of the spring is expressed by Y, and the measured data are shown in Table 1:

Table 1

Tension f (kg)

1

2

three

four

…

Spring length y(c)

…

Is the length y of the spring a function of the tension f?

Analysis: Information can be read from the table. When the tensile forces are 1kg, 2kg, 3kg and 4kg respectively, they all correspond to the length y of a spring, which satisfies the definition of the function, so the length y of the spring is a function of the tensile force f ... Usually, the first line of the function given in tabular form is the value of the independent variable, and the second line is the value of the dependent variable.

Example 2 Figure 2 shows the highest and lowest temperatures in a certain area every month of the year.

Figure 2

Figure 2 describes the relationship between which variables? Can you regard one of the variables as a function of the other?

Analysis: Three variables are given in the figure, namely the highest temperature, the lowest temperature and the month. As can be seen from the figure, the maximum and minimum temperatures change with the change of the month, and the maximum and minimum temperatures of each month are unique, so the maximum (or minimum) temperature is a function of the month. We can also find that the highest temperature in July and August is the same, which means that two independent variables correspond to the same dependent variable. Generally speaking, the horizontal axis represents the independent variable and the vertical axis represents the dependent variable for functions given in the form of images.

Example 3 Is the relationship between the following variables a functional relationship? Explain why.

(1) the relationship between the area s and the radius r of a circle;

(2) When the car is traveling at a speed of 70km/h, the relationship between the distance traveled by the car s (km) and the time spent t (hours);

(3) The area of an isosceles triangle is the relationship between its base length y (cm) and its height x (cm).

Analysis: (1) The relationship between the area s and the radius r of a circle is that when the radius is determined, the area s of the circle is also uniquely determined, so the relationship between the area s and the radius r of the circle is a functional relationship.

(2) The relationship between the distance S (km) and the time t (hour) used is that when the time t is determined, the distance s is also uniquely determined, so the relationship between the distance S (km) and the time t (hour) used is functional.

(3) The relationship between the bottom length ycm and the top height xcm is that when the top height X is determined, the bottom length Y is also uniquely determined, so the relationship between the bottom length ycm and the top height xcm is a functional relationship.

Generally speaking, the function given in the form of relation has dependent variables on the left of the equal sign and unknowns on the right of the equal sign as independent variables.

Example 4 In the following images, the one that cannot express the functional relationship is ().

Analysis: In the above four pictures, A, C and D can all represent functional relationships, because any given value of independent variable X has a unique Y value corresponding to it, but in Figure B, any given value of independent variable X has two different Y values corresponding to it, so this question should choose B.

[Question 2.9] Let M be a four-digit number less than 2006, and it is known that there is a positive integer n, so that M-n is a prime number and mn is a complete square number, and find all four-digit M that meet the conditions.

power function

The general form of power function is y = x a.

It is easy to understand if A takes a nonzero rational number, but it is not easy to understand if A takes an irrational number. In our course, there is no need to master the problem of how to understand exponential irrational numbers, because it involves very advanced knowledge of real number continuum. So we can only accept it as a known fact.

For the value of a nonzero rational number, it is necessary to discuss their respective characteristics in several cases:

First of all, we know that if a=p/q, q and p are integers, then x (p/q) = the root of q (p power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞). When the exponent n is a negative integer, let a=-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.

Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;

The possibility of 0 is ruled out, that is, for X.

The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.

To sum up, we can draw that when a is different, the different situations of the power function domain are as follows:

If a is any real number, the domain of the function is all real numbers greater than 0;

If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0.

When x is greater than 0, the range of the function is always a real number greater than 0.

When x is less than 0, only when q is odd and the range of the function is non-zero real number.

Only when a is a positive number will 0 enter the value range of the function.

Since x is greater than 0, it is meaningful to any value of a, so the following gives the respective situations of power function in the first quadrant.

You can see:

(1) All graphs pass (1, 1).

(2) When a is greater than 0, the power function monotonically increases, while when a is less than 0, the power function monotonically decreases.

(3) When a is greater than 1, the power function graph is concave; When a is less than 1 and greater than 0, the power function graph is convex.

(4) When a is less than 0, the smaller A is, the greater the inclination of the graph is.

(5)a is greater than 0, and the function passes (0,0); A is less than 0, and the function has only (0,0) points.

(6) Obviously the power function is unbounded.

Gaussian function

Let x∈R, use [x] or int(x) to represent the largest integer not exceeding x, and use non-negative pure decimal to represent x, then y= [x] is called Gaussian function, also called integer function.

Any real number can be written as the sum of an integer and a non-negative pure decimal, that is, x = [x]+(0 ≤