Calculation method:
1, reduction method
By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
For example, in the decomposition (x? +x+ 1)(x？ +x+2)- 12, which can make y=x? +x, then the original formula =(y+ 1)(y+2)- 12 =y? +3y+2- 12=y？ +3y- 10 =(y+5)(y-2) =(x？ +x+5)(x？ +x-2) =(x？ +X+5) (X+2) (X- 1)。 Example 2, (x+5)+(y-4)=8 (x+5)-(y-4)=4, x+5=m, y-4=n, the original equation can be written as m+. By using the reciprocal relationship between the definition domain and the value domain of a function and its inverse function, the value domain of the original function is obtained by finding the definition domain of the inverse function.
2. Mirror image method
According to the function image, observe the ordinate of the highest point and the lowest point.
3. Matching method
When using the collocation method of quadratic function to find the domain, we should pay attention to the range of independent variables.
4. Monotonicity method
Using the vertex or symmetry axis of quadratic function, the domain is found according to monotonicity.
5. Inverse function method
If a function has an inverse function, we can determine that its domain is the value domain of the original function by solving its inverse function.
6. Alternative methods
Including algebraic substitution and trigonometric substitution, we should pay special attention to the range of new variables after substitution.
7. Discrimination method
The discriminant method is a discriminant evaluation domain using quadratic function.
8. Composite function method
Let the compound function be f[g(x), and ]g(x) be the inner function. In order to find the range of F, we first find the range of g(x), and then regard g(x) as a whole, which is equivalent to the independent variable X of f(x), so the range of g(x) is also the definition range of f[g(x)].
9. Triangle substitution method
The basic triangular relationship is used to simplify the evaluation. For example, the square of A+the square of B = 1, the square of C+the square of D = 1, which proves that ac+bd is less than or equal to 1. It is relatively simple to directly calculate faults by triangle replacement method:
Practice: Let a=sin x, b=cos x, c=sin y, d=cos y, then AC+BD = SINX * SINY+COSX * COSY = COS(Y-X), because we know that COS (Y-X) is less than or equal to 1, so the inequality holds. ;
10, inequality method
Basic inequality method: When using a+b≥2√ab (where A, b∈R+) to find the range of function values, we should always pay attention to the conditions of inequality, that is, "one positive, two definite, three phases and so on".
1 1, separation constant method
Generally speaking, we split the molecule so that the unknown in the molecule becomes a multiple of the denominator, and then divide the constant by a formula containing the unknown.