What is the definition of a function? Urgent....

The traditional definition of a function: if there are two variables x and y in a certain change process, if y has a uniquely definite value corresponding to each definite value of x in a certain range, then y is said to be a function of x, and x is called an independent variable. We call the set of values of the independent variable x the definition domain of the function, the value of y corresponding to the independent variable x is called the function value, and the set of the value of the function is called the domain of the function.

Modern definition of a function: Let a and b be a set of non-null numbers, and f:x y is a correspondence rule from a to b, then the mapping from a to b f:

A b is called a function, denoted as y=f(x), where x a, y b, the preimage set a is called the definition domain of the function f(x), and the image set c is called the domain of the function f(x), which obviously has cb. The symbol y=f(x) is the mathematical representation of "y is a function of x", which should be understood as: x is an independent variable, it is the object to which the law is imposed; f is the law of correspondence, it can be one or several analytic formulas, it can be an image, **, or a text description; y is a function of an independent variable, when x is a certain permissible concrete value, the corresponding y value is the value of the function corresponding to the value of the independent variable, and when f is expressed by an analytic formula, the analytic formula is the analytic expression of the function.

y=f(x) is only a function symbol, not "y is equal to the product of f and x", and f(x) is not necessarily an analytic formula, in addition to the symbol f(x), g(x), g(x) and other symbols are often used to represent functions.

y=f(x) x a where x is called the independent variable x The range of values of x is the domain of a, and the value of y corresponding to x is the value of the function, and the set of function values: the range of values called f(x).

There can be a mapping definition to explain this: if a and b are two non-empty sets, and if, according to some correspondence f, for any element a in set a, there is a unique element b corresponding to it in set b, then such correspondence (including sets a, b, and the correspondence f from set a to set b) is called a mapping from set a to set b, denoted as f:a b.

where b is called the image of a under the map f, denoted as: b=f(a); A is called B with respect to mapping F's preimage. The set of images of all the elements in set a is denoted as f(a).

Function Definition: Return Type Function Name (Parameter Type 1 Parameter Name 1, ·· Parameter type n parameter name n).

Function body...

e.g. int fun(int a,int b)int c;

c=a+b;

return c; }

1. The popular meaning of a function is a relationship determined by the independent variable and the dependent variable, the independent variable may have one, two or n, but the value of the dependent variable is also the only one determined when the independent variable is determined.

2. The meaning of function is that in the field of mathematics, a function is a relation, which makes each element in one set correspond to the only element in another set.

The characteristics of the function

1. Boundedness.

Let the function f(x) be defined in the interval x, and if m>0 exists, there is always a | for all x on the interval xf（x）|m, then f(x) is said to be bounded in the interval x, otherwise f(x) is said to be unbounded in the interval.

2. Monotonicity.

Let the domain of the function f(x) be d, and the interval i is contained in d. If for any two points x1 and x2 on the interval, when x1 is called the function f(x) is monotonically increasing on the interval i; If for any two points x1 and x2 on interval i, when x1f(x2), then the function f(x) is said to be monotonically decreasing on interval i. Monotonically increasing and monotonically decreasing functions are collectively referred to as monotonic functions.

Conceptual definition of a function: given a set of numbers a, assume that the element in it is x. Now apply the corresponding rule f to the element x in a, denoted as f(x), to get another set b.

Suppose the element in b is y. Then the equivalence relationship between y and x can be expressed by y=f(x).

Function was first developed by Li Shanlan, a mathematician in the Qing Dynasty of China.

Translation, from his book Algebra

The reason for this translation is that "where there is a variable in this variable, then this is a function of the other", that is, a function refers to the change of a quantity with the change of another quantity, or that one quantity contains another quantity.

(1) Traditional definition: if there are two variables x and y in a certain change process, and for each definite value of x in a certain range, according to a certain correspondence law, y has a unique definite value corresponding to it, then y is called the function of x, x is called the independent variable, the value of y corresponding to the value of x is called the function value, and the set of function values is called the value range of the function. y is x

, which can be denoted as y

f(x) (f denotes the law of correspondence).

2) Modern definition: Let a and b be a set of non-empty numbers, and f is a correspondence rule from a to b, then a to b mapping f

A b is called a function to b, denoted as y

f(x), where x?a

y?b。The set a of the prelogue is called the domain of the function f(x), and the set c of the image is called the domain of the function f(x).

b。Note. According to the modern definition of a function, a function is a mapping between sets of numbers.

The correspondence rule f is the link between x and y, and is the core of the function, which is often expressed in an analytic formula, but in many problems, the correspondence rule f may also be inconvenient or unable to be expressed by the previous analytic formula, but in other ways (such as a number table or image, etc.). A definition field (or pre-image set) is a range of values for an independent variable, which is an indispensable component of a function, and it and the corresponding law are two important factors of a function. Functions with the same analytic expression with different domains should be treated as two different functions.

f(a) has a different meaning from f(x), f(a) is the value of the function obtained when the independent variable x a, which is a constant, while f(x) is a function of x, which is the correspondence.

Function definition: Let a and b be two sets, if according to a certain correspondence law f, for any element in set a, there is a unique element corresponding to it in set b, such correspondence is called the mapping from set a to set b, denoted as f: a-->b

When sets a and b are both sets of non-empty numbers, and each element of b has a preimage, the mapping f:a-->bIt's called a function that defines domain a to domain b

The definition in junior high school textbooks is: generally, there are two variables xy, one of which changes with the change of the other variable x, and a value of x is given and there is a unique value of y corresponding to it. x is called the independent variable and y is called the dependent variable.

FunctionsIn mathematics, a function is a relation that causes each element in one set to correspond to the unique element in another (possibly identical) set.

A dependent variable is a variable that is associated with another quantity, and any value of this quantity can find a fixed value in the other quantity.

A rule that two sets of elements of a function correspond one-to-one, and each element in the first group has only a unique counterpart in the second group.

The concept of functions is fundamental to every branch of mathematics and quantification.

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

However, functions only represent the correspondence between numbers, and mappings can also represent correspondence between points, graphs, etc. It can be said that a function is a special kind of mapping.

1. The so-called primary function is that in a certain change process, there are two variables x and y, if it can be written as y=kx+b (k is the coefficient of the primary term ≠0, k≠0, b is a constant,), then we say that y is a primary function of x, where x is the independent variable, and y is the dependent variable.

2. The basic expressions include:

Oblique truncation is more commonly used. Oblique truncation and point oblique can only be used if the slope k exists) General formula: ax+by+c=0

Oblique truncating: y=kx+b

Point slope: y-y0=k(x-x0).

Intercept formula: x a+y b = 1 (a, b are intercepts on x,y axis respectively) two-point formula: (y-y1) (x-x1) = (y2-y1) (x2-x1)3, properties (refer to oblique truncation: y=kx+b).

1) The coordinates of the intersection point of the primary function with the y-axis are always (0,b), and the intersection with the x-axis is always (-b k,0). Images of proportional functions all pass through the coordinate origin.

2) b is the intercept of the function on the y-axis, and -b k is the intercept of the function on the x-axis.

k,b determines the position of the function image:

When y=kx, y is proportional to x:

When k>0, the straight line must pass through.

1. In the third quadrant, y increases with the increase of x;

When k>0, the straight line must pass through.

In the second and fourth quadrants, y decreases with the increase of x.

y=kx+b:

When k>0,b>0, then the image of this function passes through the first.

one, two, and three quadrants;

When k>0,b>0, then the image of this function passes through the first.

1, 3, 4 quadrants;

When k>0,b>0, then the image of this function passes through the first.

1, 2, 4 quadrants;

When k>0,b>0, then the image of this function passes through the first.

Two, three, and four quadrants.

When b>0, the straight line must pass through.

1 and 2 quadrants;

When b>0, the straight line must pass through.

Three and four quadrants.

In particular, when b = 0, the straight line passes through the origin (0,0).

At this time, when k>0, the straight line only passes through the first.

3. One quadrant will not pass the first.

2. Four quadrants. When k<0, the straight line passes only the first.

The second and fourth quadrants will not pass the first.

3. One quadrant.