Problem description:

Who knows?

Excuse me.

The requirements are simple and clear.

Don't bother to make a long speech

Don't make trouble if you don't understand.

Don't copy the answers either (I'm not blind)

Analysis:

1. Univariate formula

Let f(x) be the mathematical formula, x∈D be the domain, and let y=f(x), that is, find the range of y.

(1) For very simple formulas, such as x? , |x|, √x, etc. You should learn to judge directly.

(2) Derive and judge the function values of stagnation point and boundary point.

(3) Some formulas can be transformed into equation g(x, y)=0 by inverse solution, and the value range of y can be determined by using the properties of conic curve. This method is very common in elementary mathematics, such as quadratic form, or f(x)=log[(x? + 1)/(x？ -1)] this kind of formula.

(4) For some higher-order formulas, such as f (x) = (x+1) 4+2 (x+1)? +1, which can be converted into low order by variable substitution, can be distinguished by the first two methods.

(5) Some formulas can be divided into several parts, and each part can find the value range respectively.

(6) For those that cannot be solved by previous methods, only advanced mathematical methods can be considered.

2. For multivariable formulas

This situation is generally constrained, otherwise it cannot be solved.

(1) Some cases can be transformed into univariate through constraints, such as ratio method, variable substitution and so on. For example, constraint x 2+y 2.

(2) When there are many variables, inequalities can usually be used to solve them, such as mean inequality, Cauchy inequality, rank inequality, Qin Sheng inequality, Chebyshev inequality and so on.

(3) Geometric method, many formulas can be transformed into images (circle, ellipse, hyperbola, etc. )

(3) More complicated, using advanced mathematical methods, such as Lagrange multiplier method, or solving in high-dimensional Euclidean space.