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.

A dependent variable, a function, is a variable related to another quantity, and 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.

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.

history

The mathematical term function is used by Leibniz in 1694 to describe a related quantity of a curve, such as the slope of the curve or a certain point on the curve. The function that Leibniz refers to is now called derivative function, and the functions that ordinary people except mathematicians generally come into contact with belong to this category. For a differentiable function, we can discuss its limit and derivative. Both of them describe the relationship between the change of function output value and the change of function input value, and are the basis of calculus.

In 17 18, johann bernoulli defined a function as "a function of a variable refers to a quantity composed of this variable and a constant in any way." 1748, leonhard euler, a student in johann bernoulli, said in his book Introduction to Infinite Analysis: "The function of variables is an analytical expression composed of variables and some numbers or [constants] in any way". For example, f(x) = sin(x)+x3. 1775, Euler put forward the definition of function in the book Principles of Differential Calculus: "If some quantities depend on other quantities in the following way, that is, when the latter changes, the former itself changes, then the former quantity is called the function of the latter quantity."

/kloc-mathematicians in the 0 th and 9 th centuries began to standardize all branches of mathematics. Karl Weierstrass proposed that calculus should be based on arithmetic, not geometry, so he preferred Euler's definition.

By extending the definition of function, mathematicians can study some "strange" mathematical objects, such as non-derivative continuous functions. These functions were once considered to have only theoretical value, but they were still regarded as "monsters" until the beginning of the 20th century. Later, people found that these functions played an important role in the modeling of physical phenomena such as Brownian motion.

By the end of 19, mathematicians began to try to standardize mathematics with set theory. They tried to define each mathematical object as a set. Johann peter gustav lejeune dirichlet gave the modern formal definition of function. Dirichlet's definition regards function as a special case of mathematical relationship. But for practical application, the difference between modern definition and Euler definition can be ignored.

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？ 0? 5+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? 0? 5+bx+c(a, B, C are constants, a≠0)

Vertex: y=a(x-h)? 0? 5+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？ 0? 5)/4a x 1，x2=(-b √b？ 0? 5-4ac)/2a

Three. Image of quadratic function

Do quadratic function y=x in plane rectangular coordinate system? 0? Five pictures,

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？ 0? 5)/4a ].

-b/2a=0, p is on the y axis; When δδ= b? 0? When 5-4ac=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？ 0? When 5-4ac > 0, there are two intersections between parabola and X axis.

δ= b？ 0? When 5-4ac=0, the parabola has 1 intersection points with the X axis.

δ= b？ 0? When 5-4ac < 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? 0? 5+bx+c，

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

Is that an axe? 0? 5+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 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. 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.

In short, a function is a "rule" that assigns a unique output value to each input. This "rule" can be expressed by a function expression, a mathematical relationship or a simple table listing input values and output values. The most important property of a function is certainty, that is, the same input always corresponds to the same output (note that the opposite is not necessarily true). From this point of view, a function can be regarded as a "machine" or a "black box", which converts a valid input value into a unique output value. Generally, the input value is called the parameter of the function, and the output value is called the value of the function.

The parameters and function values of the most common functions are numbers, and their corresponding relations are expressed by function expressions. The function values can be obtained by directly substituting the parameter values into the function expressions. For example,

F(x) = x2, and the square of x is the function value.

This function can also be simply extended to the case of multi-parameters. For example:

G(x, y) = xy has two parameters x and y, with the product xy as the value. Unlike before, this "rule" is related to two inputs. In fact, these two inputs can be regarded as an ordered pair (x, y), G is a function with this ordered pair (x, y) as a parameter, and the value of this function is xy.

In scientific research, there are often functions that are unknown or can't give expressions. For example, the temperature distribution at different times on the earth, this function takes the place and time of occurrence as parameters, and takes the temperature at a certain place and time as the output.

The concept of function is not limited to the calculation of numbers, or even to calculation. The mathematical concept of function is broader, which includes not only the mapping relationship between numbers. This function links the definition domain (input set) with the mapping domain (possible output set) so that each element of the definition domain uniquely corresponds to an element in the mapping domain. As described below, functions are abstractly defined as explicit mathematical relationships. Because of the generality of function definition, the concept of function is very basic for almost all branches of mathematics.