Use functional analysis methods

How to analyze function

①Patchwork: For a function analytical expression of the form f[g(x)], treat g(x) as a whole, piece the right side of the expression into the form of g(x), and then put g(x) Replace it with x and it will be ok, for example:

f(2x 1)=4x^2 2x 1，f(x):

Right side=(2x 1)^2-(2x 1) 1

∴f(x)=x^2-x 1

②Substitution method: For a function analytical expression of the form f[g(x)], let t=g(x), x can be represented by t, and pay attention to the equal definition domain, and f(t ) is ok, for example:

f[(1-x)/(1 x)]=[(1-x^2)/(1 x^2)], f(x):

Let t=(1-x)/(1 x)

Then: x=(1-t)/(1 t) (note: t≠-1)

∴ Substitute:

f(t)=2t/(t^2 1) (t≠-1)

That is: f(x)=2x/(x^2 1) (x≠-1)

③Construction method: Using the given relational expression, the variables in the relational expression can be changed to obtain a new relational expression. By solving the system of equations, the analytical expression of the function f(x) can be obtained, for example:

Suppose f(x) is a function whose domain is (0, + infinity), and f(x)=2f(1/x)√x-1 (√ is the root sign) f(x) : (The goal is to eliminate f(1/x))

Let x=1/x, get:

f(1/x)=2f(x)√(1/x)-1

Substituting it into the original equation we get:

f(x)=2[2f(x)√(1/x)-1]√x-1=4f(x)-2√x-1

∴f(x)=(2√x)/3 1/3

There is also the undetermined coefficient method, do you still want me to talk about it? So tired~~~~~

Function analysis

1. Substitution method: Given f(g(x)) and the analytical formula of f(x), the general substitution method can be used, specifically: let t=g(x), and then f(t ) can obtain the analytical formula of f(x). After the dollar exchange, the value range of the new dollar t must be determined.

Example 1. It is known that f(3x 1)=4x 3, the analytical formula of f(x).

Exercise 1. If , .

2. Matching method: Treat g(x) in the form f(g(x)) as a whole, organize it into a form containing only g(x) on the right end of the analytical expression, and then put g(x) Replace with x. Generally use the perfect square formula.

Example 2. It is known that , the analytical formula of .

Exercise 2. If , .

3. Undetermined coefficient method: given the analytical formula of the function model (such as linear function, quadratic function, exponential function, etc.), first set up the analytical formula of the function and substitute the coefficients according to the known conditions

Example 3. Suppose is a quadratic function of one variable, , and ,

and .

Exercise 3. Suppose the quadratic function satisfies , and the intercept of the image on the y-axis is 1, and the length of the line segment intercepted on the x-axis is , the expression of .

4. Method of solving the system of equations: the analytical formula of an abstract function often constructs an equation by transforming variables to form a system of equations, and uses the elimination method to solve the analytical formula of f(x)

Example 4. Suppose the function is a function defined on (-∞, 0)∪(0, ∞), and satisfies the relational expression , the analytical expression of.

Exercise 4. If , .

5. Utilize the given characteristic analytical formula: generally it is known that when x>0, the analytical formula of f(x), when x

Example 5 Suppose is an even function, when x>0, , when x

Exercise 6. For x∈R, satisfies , and when x∈[-1,0], when x∈[9,10], the expression of .

6. Inductive recursion method: Use the known recursion formula to write down several items, use the idea of ​​​​sequences to find the rules, and get the analytical formula of f(x). (General formula)

Example 6. Suppose is a function defined on , and , , the analytical formula of .

Sometimes the proof requires mathematical induction to prove the conclusion.

Exercise 5. If , and ,

Value .

Question 7. Assume , note , .

7. Related point method: Generally, set two points, one is known and one is unknown, find the connection between the two points based on the known points, represent the known points as unknown points, and finally substitute them into the known points The analytical formula can be sorted out. (Trajectory method)

Example 7: It is known that the image of the function y=f(x) and the image of y=x2 x are symmetrical about the point (-2,3), and the analytical formula of f(x).

Exercise 8. Known functions, when point P(x,y) moves on the image of y=, point Q() is on the image of y=g(x), function g(x).

8. Special value method: Generally, an abstract function about x and y is known, and an unknown number y is removed by using a special value to obtain an analytical expression about x.

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