A compound function refers to a function image in which two functions are equal to two different points in the function f(x) or the value of a function f(x), and are represented by different symbols, that is, a function image with x = f (x) and y = f (y). Compound function is often a key point in learning, because only by mastering the basic methods of compound function can we really master the meaning and properties of compound function skillfully. Only by mastering the basic method of compound function can we know how to compound the function we understand into the function we really understand. Because the compound function itself is a difficult point, we should pay attention to mastering its basic knowledge in the learning stage in order to truly master its properties. If we don't pay attention to it, we will find that the problem will not be deduced, which will lead to misunderstanding of the compound function when solving problems, and often lead to many mistakes, and the more chaotic we are, the more we regret it until the final exam can't play normally. Therefore, in the process of learning, we must pay attention to mastering the relevant knowledge of composite functions and truly master them.

1, the basic concept of composite function

According to the needs of the topic, we can separate two function images, which are called unary combination function (as shown in figure 1) and binary combination function (as shown in figure 2). Because the two functions are divided into two basic parts, the relationship between them is more complicated. For example, it is called binary combination, which is used to represent the unary combination with only two points in the function image or the combination of unary combination and double minimum value. When it is used to represent the value of x in the dual pair of f(x) and f(y) (that is, the value range of the dual logarithm of x and f(x)), it is called binary combination; For example, if the binary combination is compared to walking with hands between your legs, we call it a binary combination; For example, using unary combination to express the relationship between four variables, such as B and A, C and C, D and H, H and V, is called binary combination. Another example is to combine two parameters into a complete set of equations called binary combination and so on. After we finish the ternary combination, we only need to know what symbols are used to represent two basic concepts and what symbols are used to represent two equations with different relationships (as shown in Figure 2), then this is the ternary combination theory. Let's look at the triple synthesis problem again: cut the triple arc into several triangles, and then use the triangle area to find a triangle area, so that the triangle contains several variables.

2. Definition domain and value class of composite function

The domain of a composite function means that the function is in a certain state, including between two or more points or functions. The domain can be determined according to the known conditions, but generally it cannot exceed a certain range. Value class refers to the relationship between one-dimensional points or functions. For compound functions, the value class contains more than two points or functions with the same or different definition domains in the variable relationship. If a function is a value in a specific state, then the two value classes are related in a specific domain and have the same or different properties.

3. Classification method of composite function

According to the classification method of different symbols between different points in the function image and the classifiable types of functions represented by different symbols between them, composite functions can be divided into two categories: one is a mixed function consisting of only one parameter function and multiple parameter functions; The other is a mixed function composed of two or more parameters. According to parametric function or single parameter function, mixed function can be divided into single parameter compound (including one parameter) and multi-parameter compound (including multiple parameters). Note: The parameters used in a class of compounds and polygenic compounds are different. There is no clear definition of compounds, which can only be defined according to the least square relationship; There is also a kind of compound definition that is clear but not clear. Generally, it is only required to express the relationship between elements in a specific array, without giving the definition and quantitative relationship between elements in a specific array.

4. Mutual transformation between compound functions and functions.

The mutual transformation between compound functions is to apply the new compound function to the learning of the old function, so that the old function can correctly reflect the new compound function. When dealing with some problems, we often encounter the problem of mutual transformation between composite functions. When we encounter the problem of mutual conversion of compound functions, we should replace the original compound function with a new compound function and use the old function to find a new compound function. For example, this conversion method can interact well with new and old composite functions, and achieve the purpose of better realizing composite tasks. The compound function itself cannot directly apply the new compound value to the old compound. Only by deducing the old compound mathematical theorem can this part of compound be used correctly and effectively.