1. Concept: Let A and B be non-empty number sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B Note: y=f(x), X ∈ A.

Among them, x is called the independent variable, and the value range a of x is called the domain of the function; The y value corresponding to the value of x is called the function value, and the function value set {f(x)|? x∈A？ } is called the range of the function. Note that (1) "y = f (x)" is a function symbol, which can be represented by any letter, such as "y = g (x)"; (2) f(x) in the function symbol "y=f(x)" represents x, a function value corresponding to a number, instead of multiplying f by x.

2. The three elements of a function: definition domain, correspondence relationship and value domain.

(1) To solve all the function problems, we must carefully determine the definition domain of the function. The domain of a function includes three forms:

① Natural type: refers to the value range of meaningful independent variable x in resolution function (for example, the denominator of fractional function is not zero, the square root of even root function is not negative, and the real number of logarithmic function is positive, etc. );

(2) Restrictive type: refers to the condition of proposition or the artificial restriction on independent variable X, which is the key and often difficult point in function learning, because sometimes this restriction is hidden and easy to make mistakes;

③ Practicality: When solving the problem of function synthesis and application, the practical significance of independent variable X should be carefully examined.

(2) Finding the range of functions is a difficult mathematical problem. Middle school mathematics requires that the range of some simple functions can be found by elementary methods.

① Matching method (transforming function into quadratic function); ② Discriminant method (transforming function into quadratic equation); ③ Inequality method (using various properties of inequality); ④ Function method (using the properties of basic functions, or grasping the monotonicity of functions, function images, etc.). ).

3. The two functions are equal:

The definition of a function contains three elements, namely, the domain A, the range C and the corresponding rule F ... Two functions are the same function if and only if their domains and corresponding rules are the same respectively.

4. Interval: the classification of intervals: open interval, closed interval and half-open and half-closed interval;

5. Expression of common functions: (1) Analysis method: (2) List method: (3) Image method:

6. piecewise function: if the domain of a function is divided into several subintervals, and the analytical formula of each subinterval is different, this function is also called piecewise function;

7. compound function: if y=f(u), u=g(x), x? (a，b)，u？ (m, n), then y=f[g(x)] is called a composite function, and u is called an intermediate variable, and its value range is that of g(x).