The concept of function

The concept of (1) function

(1) is a set of non-empty numbers. If any number in a set has a unique number corresponding to it according to a certain corresponding rule, then such a corresponding relationship (including the set and the corresponding rule to) is called a function of the set, and is recorded as.

② Three elements of a function: domain, range and corresponding rules.

③ Only two domains and functions with the same corresponding rules are the same function.

(2) The concept and representation of interval

(1) Let it be two real numbers, and the set of real numbers satisfied is called a closed interval, which is recorded as; A set satisfying real numbers is called an open interval, which is recorded as; Satisfy, or the set of real numbers called semi-open and semi-closed interval, respectively; The set of satisfied real numbers is recorded as.

Note: For sets and intervals, the former can be greater than or equal to the latter.

(3) When searching for the definition domain of a function, the following principles are generally followed:

① When it is an algebraic expression, the domain is all real numbers.

When it is a fractional function, the domain is all real numbers that make the denominator not zero.

③ When the roots are even, the domain is a real number set when the modulus is non-negative.

④ The truth value of logarithmic function is greater than zero. When the base of logarithmic or exponential function contains variables, the base must be greater than zero and not equal to 1.

(5) in,.

⑥ The base of zero (negative) exponential power cannot be zero.

⑦ If it is a function composed of four operations of a finite number of basic elementary functions, its domain is generally the intersection of the domain of each basic elementary function.

⑧ For the problem of finding the domain of composite function, the general steps are: if the domain is known as, the domain of composite function should be solved by inequality.

For the function with letter parameters, find its domain and discuss the letter parameters according to the specific situation of the problem.

⑩ The definition domain of the function determined by the actual problem should not only make the function meaningful, but also conform to the actual meaning of the problem.

(4) Find the range or maximum value of the function.

The common method of finding the maximum value of a function is basically the same as the method of finding the function value domain. In fact, if there is a minimum (maximum) number in the range of a function, this number is the minimum (maximum) value of the function. Therefore, the essence of finding the maximum value of a function is the same as that of the value domain, but the angle of asking questions is different. The common methods for finding the range and maximum value of a function are:

① Observation method: For relatively simple functions, we can directly get the range or maximum value through observation.

② Matching method: the resolution function is converted into the sum of the flat modulus with independent variables and the constant, and then the range or maximum value of the function is determined according to the range of variables.

(3) Discriminant method: If a function can be transformed into a quadratic equation with coefficients, it must exist because it is a real number, so as to determine the range or maximum value of the function.

④ Inequality method: Using basic inequality to determine the range or maximum value of the function.

⑤ Substitution method: Through variable substitution, complexity and difficulty can be simplified. Trigonometric substitution can transform the maximum problem of algebraic function into the maximum problem of trigonometric function.

⑥ Inverse function method: Determine the value range or maximum value of a function by using the reciprocal relationship between the definition range and the value range of the function and its inverse function.

⑦ Number-shape combination method: Determine the range or maximum value of a function by using function images or geometric methods.

Monotonicity method of functions.

Representation of function

(5) Representation method of functions

There are three commonly used methods to express functions: analytical method, list method and image method.

Analytic method is to express the corresponding relationship between two variables with mathematical expressions. List method is to express the corresponding relationship between two variables by list. Image method is to use images to express the corresponding relationship between two variables.

(6) The concept of mapping

(1) Let sum be two sets. If any element in a set has a unique element corresponding to it according to certain corresponding rules, then such a corresponding relationship (including the set and the corresponding rule to) is called the mapping from the set to, and is recorded as.

② Given a set-to-set mapping, and. If an element corresponds to an element, then we call it the image of the element, and the element is the original image of the element.

Basic properties of functions

Monotonicity and Maximum (Minimum) Value

Monotonicity of (1) Function

① Definition and determination method

functional

nature

definition

picture

Judgment method

functional

monotonicity

If the values of any two independent variables belonging to an interval in the domain I are x 1 and x2, when X 1

(1) utilization definition

(2) Using the monotonicity of known functions

(3) Use function diagram (in interval diagram)

E.g., rising to increase)

(4) Using compound function

If the values of any two independent variables belonging to an interval in the domain I are x 1 and x2, when X 1

(1) utilization definition

(2) Using the monotonicity of known functions

(3) Use function diagram (in interval diagram)

Decrease as follows)

(4) Using compound function

(2) In the public domain, the sum of two increasing function is increasing function, the sum of two subtraction functions is subtraction function, increasing function minus one subtraction function is increasing function, and subtraction function minus one increasing function is subtraction function.

(3) For the compound function, make, if it is incremental, if it is incremental, is incremental; If it decreases, it decreases, if it decreases, it increases; If it is an increase, it is a decrease, which is a decrease; If it is a decrease, it is an increase, and it is a decrease.

(2) Images and attributes of the "check" function

Increasing function is above and is above and minus function is above.

(3) the definition of the maximum (minimum) value

① Generally, let the domain of a function be, if a real number satisfies: (1) for any; (2) Existence makes existence. Then, we call it the maximum value of the function, and write it as.

② It is generally assumed that the domain of a function is, if a real number satisfies: (1) for any; (2) Existence makes existence. Then, we call it the minimum value of the function, and write it as.

Second, parity.

(4) Functional equivalence

① Definition and determination method

functional

nature

definition

picture

Judgment method

functional

odevity

If any x in the domain of function f(x) has f (-x) =-f(x), then function f(x) is called odd function.

(1) Use definition (it is necessary to judge whether the domain is symmetrical about the origin first)

(2) Use an image (the image is symmetrical about the origin)

If any x in the definition domain of the function f(x) has f (-x) = f(x), the function f(x) is called an even function.

(1) Use definition (it is necessary to judge whether the domain is symmetrical about the origin first)

(2) Use an image (the image is symmetrical about the Y axis)

(2) if the function is odd function, defined in, then.

③ The symmetry intervals of odd function on both sides of the axis are the same, and the symmetry intervals of even functions on both sides of the axis are opposite.

④ In the public domain, the sum (or difference) of two even functions (or odd function) is still an even function (or odd function), the product (or quotient) of two even functions (or odd function) is an even function, and the product (or quotient) of an even function and a odd function is odd function.

[Supplementary knowledge] Functional image

(1) drawings

Drawing by tracing points;

(1) Determine the functional domain; ② Analytical resolution function;

③ Discuss the properties of functions (parity and monotonicity); ④ Draw the image of the function.

Drawing with the transformation of basic function image;

It is necessary to accurately remember the images of various basic elementary functions such as linear function, quadratic function, inverse proportional function, exponential function, logarithmic function, power function and trigonometric function.

① Translation and transformation

② Telescopic transformation

③ Symmetric transformation

(2) Look at the map

For the image of a given function, we should be able to study the definition range, range, monotonicity and parity of the function from the left and right, up and down range, changing trend and symmetry of the image, and pay attention to the relationship between the image and the parameters in the resolution function.

(3) Use charts

The image of function vividly shows the essence of function, which provides a "shape" intuition for studying quantitative relations. It is an important tool to explore ways to solve problems and obtain results. Pay attention to the thinking method of combining numbers and shapes when solving problems.

Several common methods in the field of evaluation

(1) collocation method: the common collocation method of "quadratic function" function, such as finding the function, can be transformed into solving it.

(2) Basic function method: Some functions composed of basic functions can be solved by the range of basic functions, for example, functions can be solved by the range of function sum.

(3) Discriminant method: through the discriminant evaluation domain of the real root of quadratic equation. If you want to find the range of a function

By, if, then, so it is a value in the function value domain; If so, it will be obtained, so the range of values sought is

(4) Separation constant method: it is often used to find the range of "fractional" function. If you want to find the range of a function, because

And, so, so.

(5) Using basic inequalities to evaluate the range, such as finding the range of functions.

At that time,; At that time, if, then

If, then, the required scope is

(6) Use the monotonicity of the function to find the evaluation domain, such as finding the range of the function.

Therefore, functions are decreased, increased, decreased and increased in, so that the required range can be obtained as follows.

(7) Image method: If the image of the function is easy to make, the range of the function can be found intuitively according to the image (this method is often used to find the range of some piecewise functions).

Concepts of function and mapping