Function thought refers to analyzing, reforming and solving problems with the concept and nature of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are mutually transformed and interrelated, thus solving problems.

Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; The inequality problem is also closely related to the fact that the equation is a close relative. There is no essential difference between functions and multivariate equations. For example, the function y = f (x) can be regarded as a binary equation f (x)-y = 0 about x and y, so it can be said that the learning of functions is inseparable from formulas. Column equation, solving equation and studying the characteristics of equation are all important considerations when applying the idea of equation.

Function describes the relationship between quantities in nature, and the function idea establishes the mathematical model of function relationship by putting forward the mathematical characteristics of the problem, so as to carry out research. It embodies the dialectical materialism view of "connection and change". Generally speaking, the idea of function is to use the properties of function to construct functions to solve problems. Commonly used properties are monotonicity, parity, periodicity, maximum and minimum, image transformation and so on. What we are required to master skillfully are the specific characteristics of linear function, quadratic function, power function, exponential function, logarithmic function and trigonometric function. In solving problems, it is the key to use the thought of function, be good at digging the implicit conditions in the problem, construct the analytical formula of function and skillfully use the properties of function. Only by in-depth, full and comprehensive observation, analysis and judgment of a given problem can we have a trade-off relationship and build a functional prototype. In addition, equation problems, inequality problems and some algebraic problems can also be transformed into related functional problems, that is, solving non-functional problems with functional ideas.

Function knowledge involves many knowledge points and a wide range, and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination. The common types of questions we use function thought are: when encountering variables, construct function relations to solve problems; Analyze inequality, equation, minimum value, maximum value and other issues from the perspective of function; In multivariable mathematical problems, choose the appropriate principal variable to reveal the functional relationship; Practical application of questions, translation into mathematical language, establishment of mathematical models and functional relationships, and application of knowledge such as functional properties or inequalities to answer questions; Arithmetic, geometric series, general term formula and sum formula of the first n terms can all be regarded as functions of n, and the problem of sequence can also be solved by function method.

Equivalent transformation

Equivalence transformation is an important thinking method to transform the problem of unknown solution into a problem that can be solved within the existing knowledge. Through continuous transformation, unfamiliar, irregular and complex problems are transformed into familiar, standardized and even simple problems. Over the years, the idea of equivalent conversion has been everywhere in the college entrance examination. We should constantly cultivate and train our consciousness of transformation, which will help to enhance our adaptability in solving mathematical problems and improve our thinking ability and skills. There are equivalent transformations and unequal transformations. Equivalence transformation requires that causality in the transformation process is sufficient and necessary to ensure that the result after transformation is still the result of the original problem. The full or necessary transformation of the non-equivalent process and the necessary revision of the conclusion (such as the root test of unreasonable equations) can bring people a bright spot of thinking and find a breakthrough to solve the problem. In application, we must pay attention to the different requirements of equivalence and non-equivalence, and ensure its equivalence and logical correctness when realizing equivalence transformation.

C.A. Yatekaya, a famous mathematician and professor at Moscow University, once said in a speech entitled "What is a problem-solving" to mathematical Olympians: "Solving a problem is to turn it into a solved problem". The process of solving mathematical problems is the transformation from unknown to known, and from complex to simple.

The equivalent transformation method is flexible and diverse. There is no unified model for solving mathematical problems with the idea of equivalent transformation. It can be transformed from number variables, from shape to number; Equivalent conversion can be carried out at the macro level, such as the translation from ordinary language to mathematical language in the process of analyzing and solving practical problems; It can realize transformation within the symbol system, which is called identity deformation. Elimination, substitution, combination of numbers and shapes, and evaluation domain all embody the idea of equivalent transformation. We often carry out equivalent transformation among functions, equations and inequalities. It can be said that the equivalent transformation is to raise the algebraic deformation of identity deformation to keep the truth of the proposition unchanged. Because of its diversity and flexibility, we should reasonably design the ways and methods of transformation to avoid copying the questions mechanically.

When implementing equivalent transformation in mathematical operations, we should follow the principles of familiarity, simplification, intuition and standardization, that is, we should turn the problems we encounter into problems we are more familiar with; Or turn more complicated and tedious problems into simpler ones, such as from transcendence to algebra, from unreasonable to rational, from fractions to algebraic expressions and so on. Or more difficult and abstract problems are transformed into more intuitive problems to accurately grasp the problem-solving process, such as the combination of numbers and shapes; Or from nonstandard to standard. According to these principles, mathematical operations can save time and effort in the process of transformation, such as pushing the boat with the current, often infiltrating the idea of equivalent transformation, which can improve the level and ability of solving problems.

Classified discussion

When solving some math problems, sometimes