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Theoretical basis of mathematical analysis 3: the concept of function
Given two sets of real numbers D and M (usually replaced by R), if there is a corresponding rule F, so that each number X in D has a unique number corresponding to it, it is said that F is a function defined on number set D.

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Call x the independent variable and y the dependent variable.

The number set D is called the domain of function F, and the number Y corresponding to X is called the function value of F at point X, which is often recorded as f(x).

The set of all function values is called the range of function F.

1. The domain D and the corresponding rule F are two main factors that determine a function, and they are often used to represent a function.

The two functions are the same, that is, they have the same fields and corresponding rules.

2. Existence domain: analytic method (formula method) represents a function, and the definition domain of the function often takes all the meaningful independent variable values of the operator, which is called existence domain.

At this time, the domain d of the function can be omitted, and the function can only be represented by the corresponding rule f, which is called "function y=f(x)" or "function f"

3. The function f gives the single-valued correspondence between the point set d on the X axis and the point set m on the Y axis, which is also called mapping.

For f(a), it is called the image of A under the mapping F, and A is called the original image of f(a).

4. Each one can only have a unique Y value corresponding to it, and the function defined in this way is called a single-valued function.

If the same x value can correspond to multiple y values, a function is called a multivalued function.

Example: Symbolic Function

For example: f(x)=|x|

A function is represented by an ordered set of pairs of numbers.

On the coordinate plane, each element (x, y) of set G represents a point on the plane, and set G depicts the image of this function on the coordinate plane.

Dirichlet function:

Riemann function:

Given the sum of two functions, remember and set it.

If the x value of g(x)=0 is excluded from d,

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Note: If is, four operations cannot be performed.

There are two functions

Remember, if, for each, the function defined on can be determined by the function G corresponding to the unique value U and the function F corresponding to the unique value Y in D, where X is the independent variable and Y is the dependent variable.

Compound function of f and g;

Write down or

F is an external function, g is an internal function and u is an intermediate variable.

Note: F and G can be combined if and only if.

Let the function satisfy:

For each value y in the interval f(D), only one value x in d makes f (x) = y.

Then according to this correspondence rule, a function defined on f(D) is obtained, which is called the inverse function of f.

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1.f has an inverse function, then f is a one-to-one mapping between d and f(D), which is called the inverse mapping of mapping f, each f(a) in f(D) corresponds to a unique a in d, a is called the image of f(a) under the inverse mapping, and f(a) is called the original image of a under the inverse mapping.

2.f sum is inverse function.

3. inverse function habit notation: x is the independent variable and y is the dependent variable.

Constant function:

Power function:

Exponential function:

Logarithmic function:

Trigonometric function:

Inverse trigonometric function:

Given a real number, specify

A function obtained by finite four operations and compound operations of basic elementary functions.

Functions that are not elementary functions are called non-elementary functions.