The idea of isomorphism is to construct both sides of inequality into algebraic expressions with the same structure, and then solve inequality with monotonicity of function.

Isomorphism is very popular in recent college entrance examination questions, such as 2020 national volume 1 theory 10, literature 10, 2020 new college entrance examination Shandong volume 22. ...

If time can stay at the moment when we first met.

Let's start with an example:

So the mysterious isomorphism is that simple? We divide different variables into two sides of inequality, and both sides naturally become the same form. Wit! The technique of isomorphism is therefore easy to find.

Yes, learning math well means constantly summing up the rules in solving problems, so that you can be confident when you face similar problems next time.

Well, you think you've learned. Are you ready for fun?

No, sir, don't leave in a hurry. It's more than that. Let's read on.

Let's look at another example:

My first reaction to this problem is: solving inequalities, which I am good at. I'll find the denominator first, and then move and merge the terms to find the root formula bolabola…… ... ...

Hey, wait, why did this project appear six times? That's out of line! ? It's so complicated, you want me to learn the competition! ! ?

Haha, use isomorphism! Look at our solution.

What? Let me have a closer look. I seem to have found something amazing. Inequality can be solved like this! ? Both sides of this inequality are the same variables. The idea of isomorphism guides us that both sides should become the same structure, and the function represented by this structure should be monotone, so that we can solve the inequality through monotony.

How's it going? Is it cool? Do you still think you can solve it mechanically? So the idea of isomorphism needs to deeply analyze the characteristics of algebra and our flexible algebraic deformation ability. What do you think? Change an inequality (equation) until suddenly a moment, a formula with neat edges pops onto the paper, which is the time when the beauty of mathematics and thinking blooms.

Let's summarize the steps of solving inequality isomorphism.

1, which transforms both sides of inequality into algebraic expressions with the same structure;

2. Find the generating function and determine the monotonicity of the generating function;

3. Solve the variable value domain with monotonicity.

Suddenly looking back, she was in the dim light.

Isomorphism is the ultimate test form of functional modules in a certain direction in college entrance examination, but it is not a flower growing on a cliff, and its roots actually lie in our classroom and homework practice. It is no exaggeration to say that monotonicity is the core of a function. Many topics in functions and derivatives revolve around monotonicity. The topic of isomorphism is derived from the most basic topic of monotonicity. Look:

I don't understand question 4! Do you understand, my friends? Many colorful representations are rooted in the basic soil, and isomorphism is no exception. It is not mysterious, but it skips the middle step of evolution. What we examine is to see how deeply you understand the basic knowledge and basic questions.

Isomorphism is not only applied to solving inequalities, but also has made great achievements in dealing with inequalities, proving inequalities and solving equations.

How's it going? Have you ever marveled? See if there is a strong impulse to explore isomorphism here?

Wan Li has a long way to go, so be ready to start again.

Readers with derivative basis can continue reading.

Derivative is a powerful weapon to study monotonicity of functions. If the initial contact function was in the Stone Age, the appearance of derivative directly brought us to the steam engine age. With the derivative, my mother no longer has to worry that I can't handle the monotonicity of a disgusting function. Using isomorphism in the last derivative.

This is the power of isomorphism in solving parameter problems in derivative modules. Many readers may have questions about the isomorphic deformation above. I want to know by what magical method did you get there, or were you trying at random? Spoiler: The isomorphic deformation above is called Lambert isomorphism, which is an isomorphic form with very high frequency in the college entrance examination. This isomorphic deformation is very regular, and we have also summarized a set of operational steps.