How to understand the substitution method in high school function, for example

How to understand the substitution method in high school function, for example

When solving mathematical problems, a formula is regarded as a whole and replaced by a variable to simplify the problem. This is called substitution method. The essence of substitution is transformation, and the key is to construct and set elements. The theoretical basis is equivalent substitution. The purpose is to transform the research object and move the problem to the knowledge background of the new object, so as to standardize the non-standard problems and simplify the complex problems, Make it easy to handle
The substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, the scattered conditions can be connected, the implied conditions can be revealed, or the conditions can be connected with the conclusion, or the complex calculation and deduction can be simplified in a familiar form
It can transform high order into low order, fraction into integral, irrational into rational, transcendental into algebraic. It has a wide range of applications in the study of equations, inequalities, functions, sequences, trigonometry and other problems
The methods of substitution include: local substitution, trigonometric substitution, mean substitution, etc. local substitution, also known as global substitution, is to replace an algebraic expression with a letter when it appears several times in the known or unknown, so as to simplify the problem. Of course, sometimes it can only be found by deformation. For example, to solve the inequality: 4 + 2-2 ≥ 0, we first change it to let 2 = t (T > 0), It becomes the familiar problem of solving quadratic inequality and exponential equation
Trigonometric commutation is used to remove the radical sign or transform to trigonometric form. When it is easy to find, it mainly uses the known algebraic formula and trigonometric knowledge to exchange. For example, when finding the value range of function y = +, it is easy to find x ∈ [0,1], let x = sin α, α∈ [0,]. The problem turns into the familiar problem of finding the value range of trigonometric function, If the variables X and y are suitable for the condition x + y = R (r > 0), then the trigonometric substitution x = RCOs θ and y = rsin θ can be made into trigonometric problems
If x + y = s, let x = + T, y = - t, etc
When we use the substitution method, we should follow the principle that is conducive to operation and standardization. After substitution, we should pay attention to the selection of the range of variables. We must make sure that the range of new variables corresponds to the value range of the original variables and cannot be narrowed or expanded
You can observe the formula first, and you can find that there are always the same formula in the formula of the substitution method, and then replace them with a letter to calculate the answer. If there is a letter in the answer, you can bring the formula in and calculate it