Skills of solving problems with quadratic resolution function.
Common solutions to analytic functions;
(1) undetermined coefficient method: (known function types, such as linear function, quadratic function, inverse proportional function, etc. ): If the structure of f(x) is known, you can set an expression with parameters, and then list the equations or equations according to the known conditions, so as to find the undetermined parameters and get the expression of f(x). The undetermined coefficient method is an important mathematical method, which is only suitable for solving the analytical formula of known functions.
(2) Substitution method (pay attention to the range of Singapore dollar): We know the expression of f(g(x)). If f(x) is required, we always set t=g(x), so as to get the expression of x = (g (- 1)) (t), and then substitute it into f(g(x).
(3) Matching method (whole replacement method): If the expression of f(g(x)) is known, it is difficult to find the expression of f(x) by replacement method (for example, g(x) has no inverse function). You can regard g(x) as a whole, replace the right side with a formula composed of g(x), and then substitute it to get f(x).
(4) Elimination method (e.g. the independent variables are reciprocal, f(x) is known as odd function, and g(x) is an even function, etc. ): If the form of an equation with a function as an element is known, if you can try to construct another equation to form an equation group, and then solve this equation group to find the function element, this method is called elimination method.
(5) Assignment method (special value substitution method): When finding the expressions of some functions or finding some function values, sometimes some variables in known conditions are assigned to make the problem simple and clear, so it is easy to find the expressions of functions.
Finding resolution function is an important content of middle school mathematics and one of the important test sites of college entrance examination. Geek mathematics helps to give the basic method of finding the resolution function for teachers and students' reference.
First, the definition method
The method of finding its analytic expression according to the definition of function.
Second, alternative methods.
When using method of substitution to solve the resolution function, we must consider the range of "yuan", that is, the domain of f(x).
Third, the equation method
According to the meaning of the question, the method of solving analytic function by equation is established.
The key to solving analytical formula by equation group method is to construct another equation according to the characteristics of formulas in known equations.
Fourth, professional methods.
The method of finding the resolution function by taking the special value of a variable.
Verb (verb abbreviation) undetermined coefficient method
Given the type of resolution function, we can set the form of its analytical formula, and establish the equation of undetermined coefficient according to the known conditions, so as to find out the method of resolution function.
Functional attribute method of intransitive verbs
The method of finding the resolution function by using the parity, monotonicity and periodicity of the function.
Seven, inverse function method
The method of finding the analytic expression of inverse function by using the definition of inverse function.
Eight, "instant definition" method
In this paper, an "immediate definition" function is given, and the method of solving the resolution function according to this definition is given.
Nine, modeling methods
The method of establishing function model according to practical problems.
Mirror image method
The method of finding analytic expression by using function image.
XI。 Trajectory method
Set any point P(x, y) on the function image, and establish the equation about x and y according to the meaning of the question, so as to find out the method of distinguishing the function.
practise
1. It is known that the vertex of the image of quadratic function is (-2,3), which passes through (-1, 5). Find the analytic expression of this quadratic function.
2. Given that the image of a quadratic function intersects the X axis at points (-2,0) and (4,0), and the maximum value is -4.5, find the analytic expression of this quadratic function. 3. It is known that the two intersections of the quadratic function f(x) and the X axis are (-2,0) and (3,0), and f(0)=-3, so find f(x).
4. It is known that f(x) is a linear function, which satisfies 3f(x+1)-2f (x-1) = 2x+17 to find f (x).
5. It is known that the quadratic function f(x) satisfies: f(x+1)+f (x-1) = 2x2-4x, so find f (x).
6. Given that f(x) is a linear function and f[f(x)]=9x+8, find f(x).
7. Given that f (x) = x 2-1,find f (x+x 2).
8. It is known that the function f(x) satisfies: f(x)-2f(-x)=3x+2, and find f(x).
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