A compulsory function of senior one mathematics and its representation 1 a compulsory function of senior one mathematics and its representation;

Function and its representation

The detailed knowledge document contains the concept of function, mapping, judging principle of function relationship, function interval, three elements of function, definition domain of function, function value of concrete or abstract numerical value, function value domain, expression method of function, etc.

The screenshot of the first page of the document is as follows:

1。 The difference between function and mapping:

2。 Find the domain of function

The domain of the common analytic function f(x) can be summarized as follows:

① When f(x) is an algebraic expression, the domain of the function is r.

② When f(x) is a fraction, the domain of the function is a set of real numbers that make the denominator of the fraction non-zero.

(3) When f(x) is an even root, the domain of the function is a group of real numbers whose root number is not less than 0.

(4) When f(x) is logarithmic, the domain of the function is the set of real numbers that make the real number positive and the base is positive instead of 1.

⑤ If f(x) consists of several mathematical expressions, then the function domain is the set of real numbers that make all the expressions meaningful, that is, the intersection of the set of real numbers that make all the expressions meaningful.

⑥ The domain of compound function is the intersection of the domain of basic function.

⑦ For the function determined by the background of the actual problem, its definition domain is limited by the actual problem in addition to the above.

3。 Find function range

(1), observation method: by observing the definition and properties of the function, combined with the analytical formula of the function, the range of the function is obtained;

(2) Matching method; If a function is a quadratic function or can be written in the form of a quadratic function by method of substitution, then the range of the function can be found by the formula on the right side of the function through the range of independent variables;

(3), discrimination method:

(4) Number-shape combination method; By observing the image of the function, the range of the function is obtained by combining numbers and shapes.

(5) Alternative methods; Replace some quantities in the function formula with new variables, so that the function can be transformed into a function form with new variables as independent variables, and then the range can be obtained;

(6) Using monotonicity of functions; If the function is strictly monotonous in a given domain interval, then the range can be obtained by using the function value of the endpoint;

(7) Using basic inequalities: For some special fractional functions and functions higher than quadratic, we can use important inequalities to find the range of functions;

(8) Maximum method: For the continuous function y=f(x) on the closed interval [a, b], the extreme value of y=f(x) on the interval [a, b] can be found and compared with the boundary value f(a). F(b), find the maximum value of the function and get the range of the function y;

(9) Inverse function method: If a function has an inverse function in its definition domain, then finding the value domain of the function can be transformed into finding the definition domain of the inverse function.

A compulsory function of senior one mathematics and the representation of its knowledge points.

The knowledge in this section includes monotonicity, parity, periodicity, maximum, symmetry and images of functions. Monotonicity, parity, periodicity, maximum and symmetry of functions are the basis of learning function images, and function images are their synthesis. So understand the previous knowledge points, and the image of the function will be solved.

First of all, the monotonicity of the function

1, the definition of monotonicity of function

2. Judgment and proof of monotonicity of function;

(1) definition method

(2) Complex variable function analysis method

(3) Derivative proof method

(4) Image method

Second, the parity and periodicity of the function

1, the definition of parity and periodicity of function

2. Methods to judge and prove the parity of functions.

3. The method of judging the periodicity of the function

Third, the function of image.

1, the method of function image

(1) stippling

(2) Image transformation method

2. Image transformation includes images: translation transformation, expansion transformation, symmetry transformation and folding transformation.

Common inspection methods

This part is an indispensable part of Duan and the college entrance examination, and it is the focus and difficulty of Duan and the college entrance examination. There are multiple-choice questions, fill-in-the-blank questions and solutions, and the questions are more difficult. In solving problems, we can combine each chapter of high school mathematics, mostly advanced questions. More attention should be paid to the monotonicity, maximum and image of the function.

Misunderstanding reminder

1. To find the monotone interval of a function, the domain of the function is required first, that is, the principle that the domain of the function problem takes precedence is followed.

2. Monotone interval must be expressed by interval, not by set or inequality. Monotone interval is generally written as an open interval, regardless of the endpoint problem.

3. Multiple monotonous intervals cannot be connected by "or" and ","and can only be separated by commas.

4. To judge the parity of a function, we should first consider the domain of the function. If the domain of a function is not symmetric about the origin, then the function must be a odd function or even function.

5. As a function, it is generally to simplify the analytical formula first, and then determine the image as a function by tracing points or image transformation.