What are abstract functions and their applications

We call functions that do not give a concrete analytic formula an abstract function. Since this kind of problem can comprehensively test students' understanding of the concept and properties of functions, and at the same time, abstract function problems integrate the definition domain, value range, monotonicity, parity, periodicity and image of the function The basis for solving abstract function problems is to be familiar with the basic knowledge of functions. If you don't even have a basic knowledge of functions, solving abstract function problems can only be empty talk.

Specifically, to learn functions well, you need to grasp the properties of common functions. For example, the properties of functions involved in secondary schools are generally monotonicity, parity, boundedness, and periodicity. Common functions include exponential functions, logarithmic functions, trigonometric functions, quadratic functions, tick functions (y=x+a x(a>0)), and so on.

To put it simply, an abstract function is a function that does not give a concrete analytic formula. Application is related to real life and production. Applications in mathematics are often realized in the form of word problems, do you understand?

We call this kind of function an abstract function that does not give a specific analytic expression of the function, but gives some properties or corresponding conditions of the function.

Conclusion 1: (One point symmetry) If the function y=f(x), for any, satisfies or, then the image of the function y=f(x) is symmetrical with respect to (a,0).

Corollary: If the function y=f(x) satisfies any of the conditions, then the image of the function y=f(x) is symmetrical with respect to (,0).

Conclusion 2: (two-point symmetry) If the image of the function y=f(x) is symmetrical with respect to both point (a,0) and point (b,0) symmetry (where a≠b), then y=f(x) is a periodic function, period.

Proof : The function y=f(x) is symmetrical with respect to both point (a,0) and point (b,0) symmetry.

y=f(x) is the periodic function, period.

Corollary: The function y=f(x) is an odd function, and its image is symmetrical (a≠0) with respect to the point (a,0), then the function y=f(x) is t=2|a|The periodic function of .

Conclusion 3: (Axisymmetry) If the function y=f(x) satisfies any OR for any pair, then the function y=f(x) is symmetric with respect to x=a.

Corollary: If the function y=f(x) satisfies the condition for any x, then the image of the function y=f(x) is symmetrical with respect to the straight line.

Conclusion 4: (Axisymmetric Symmetry) If the image of the function y=f(x) is symmetrical with respect to both the line x=a and the straight line x=b symmetrical (a≠b), then the function y=f(x) is t=2|b－a|The periodic function of .

Proof : y=f(x) symmetry with respect to x=a and x=b.

t=2|b－a|is the period of y=f(x).

Conclusion 5: (Point axis symmetry) If the image of the function y=f(x) is symmetrical with respect to both the point (a,0) and the line x=b symmetrical (where a≠b), then the function y=f(x) is the period t=4|b－a|The periodic function of .

Proof: The function y=f(x) is symmetrical with respect to points (a,0) and symmetry with respect to x=b.

is the period of y=f(x).

Special case 1: If the odd function y=f(x) image is symmetrical with respect to the straight line x=b(b≠0), then the function y=f(x) is periodic t=4|b|The periodic function of .

Special case 2: If the even function y=f(x) image is symmetrical (a≠0) with respect to the point (a,0), then the function y=f(x) is periodic t=4|a|The periodic function of .

The properties of abstract functions are periodicity, symmetry, symmetry points, etc.

1. Periodicity.

If an abstract function satisfies f(x+a)=f(x) or f(x-a)=f(x) (where a>0) is constant, then the function is a periodic function with a period of 2a.

2. Symmetry.

If the image of an abstract function is symmetrical with respect to the lines x=a and x=b, then the function is a periodic function with a period of 2|a-b|。

3. Symmetry point.

If the image of an abstract function is symmetrical with respect to the points (a,0) and (b,0), then the function is a periodic function with a period of 2|a-b|。

Significance of learning mathematics:

1. Improvement of thinking ability.

Shuwei ethnology is not only a kind of knowledge, but also a way of thinking, including Tuidongshan group theory, logic, proof, induction, analogy, etc. Studying mathematics can help us develop these thinking skills, enhance our analytical and problem-solving skills, and can also help us better understand other subjects.

2. Insight into the world.

Mathematics is an insight into the world that can help us understand the nature and laws of things. For example, studying geometry can help us understand the basic concepts of space and shape, and studying statistics can help us understand the laws and meanings of data.

3. Stimulate interest and curiosity.

There are many interesting and magical phenomena and problems in mathematics, such as mathematical paradoxes, mathematical conjectures, geometric diagrams, etc. Learning mathematics can spark our interest and curiosity in mathematics and science, pushing us to keep exploring and discovering new knowledge.

An abstract class is a class with a pure virtual function that cannot create an object, but can only declare pointers and references, and is used for interface declarations and runtime polymorphism of the underlying class. In addition, if a derived class of an abstract class is not implemented in the process of backtracking to the root of the inheritance system, the class is also an abstract class and cannot create objects.

The function of the virtual function is to realize dynamic concatenation, that is, to dynamically select the appropriate member function in the running stage of the program, after the virtual function is defined, the virtual function can be redefined in the derived class of the base class, and the function redefined in the derived class should have the same number of parameters and the type of parameters as the virtual function. In order to achieve a unified interface, different definition processes. If the virtual function is not redefined in the derived class, it inherits the virtual function of its base class.

Let x=1 y=1

f(1*1)=f(1)+f(1)

f(1)=o

Let x=1 y=1

f(-1*(-1))=f(-1)+f(-1)=02f(-1)=0 f(-1)=0 y=-1

f(-x)=f(-1*x)=f(-1)+f(x)=f(x), so it is an even function.

Let y=1 x

f(x*1 x)=f(x)+f(1 x)=f(1)=0, then f(1 x)=-f(x).

If x1x1

Then x2 x1>1

So f(x2 x1) >0

So f(x2)-f(x1)>0

So it's an increment function.

Solution: The first question: take x=y=1, substitute f(xy)=f(x)+f(y) to obtain, f(1)=f(1)+f(1), so f(1)=0

The second question: take y=x, substitute f(xy)=f(x)+f(y), f(x)=2f(x), then f(x) = f(x), f(-x) = f(-x) )=f(x)=f(x), f(x) is an even function.

Question 3: Let a b 0 and set a=b+c, where c 0, then f(a)=f(b)+f(c), that is, f(a)-f(b)=f(c) 0, that is, it proves that f(x) is an increasing function on (0,+).

Grasp the definition of the function (the characteristics of the function of increment and decrease, parity, special values, extreme values, etc.).

We call functions that do not give a concrete analytic formula an abstract function. Since this type of problem can comprehensively test the student's understanding of the concept and properties of functions, the abstract function problem also integrates the definition domain, value range, monotonicity, parity, periodicity, and image of the function.

Interpret abstract functions.

For f(x), the bridge range of (x) = the defined domain of f(x).

f: denotes the same operation: f(x) is equivalent to f[g(x)], and x) is in the same range as [g(x)].

For f(x+1), the range of (x+1) is not equal to the defined domain of f(x+1).

For f(x) vs. f(x+1): where (x) is equal in range to (x+1), 1Knowing the domain of the function f(x), find the domain of destroying the domain of the fierce f[g(x)].

If f(x) defines the domain as: a