College entrance examination requirements

Understand the concept of functional parity, master some simple methods to judge functional parity, master the definition and image characteristics of functional parity, judge and prove functional parity, and use functional parity to solve problems.

Knowledge point induction

1 definition of function parity;

2 the nature of the parity function:

The (1) domain is symmetric about the origin; (2) The image of even function is symmetrical about the axis, and that of odd function is symmetrical about the origin;

3 is an even function.

4 If the domain of odd function contains, then

To judge the parity of a function, we must first study the domain of the function, and sometimes we can simplify the function formula, but we must pay attention to keeping the domain unaffected;

6. Keeping in mind the image characteristics of the even-odd function is helpful to judge the parity of the function;

7 The parity of the judgment function can sometimes be defined in equivalent form:

,

8 Assuming that the domains are respectively, then on their common domains:

Odd+odd = odd, odd = even, even+even = even, even = even, odd = odd.

1 The parity of a function must be judged according to the definition of parity of the function. In order to facilitate the judgment, the equivalent form of the definition is often adopted: f (? x)=？ f(x)？ f(？ x)f(x)= 0；

The premise of discussing the parity of function is that the domain of function is symmetrical about the origin, which should be paid attention to;

3 If the domain of odd function contains 0, then f(0)=0, so "f(x) is odd function" is an insufficient and unnecessary condition for "f(0)=0";

Odd function's image is symmetrical about the origin, and even function's image is symmetrical about the Y axis, so we can judge the parity of the function according to the symmetry of the image.

5 If there is a constant t, so that f(x+T)=f(x) holds for any x in the domain of f(x), then t is called the period of function f(x). Generally speaking, period means that the domain of the minimum positive period of a function must be an infinite set.

To understand the definition of functional parity, we should not only stay on the two equations of f(-x)=f(x) and f(-x)=-f(x), but also make it clear that any x in the domain has f(-x)=f(x) and f(-x)=-f(x).

The difficulty in this part is the comprehensive application of monotonicity and parity of functions. Students are required to mobilize relevant knowledge and choose appropriate methods to solve problems according to known conditions.

(5) the periodicity of the function

Definition: If t is a non-zero constant, the constant holds for any x in the definition domain.

Then f(x) is called a periodic function, and t is called the period of this function.

Example: (1) If the function is odd function on R and increasing function on R, and

Then ① About symmetry; ② The period is:

③ In (1, 2) is a function (increase or decrease);

④ =, then

(2) Let it be a periodic function defined on, with a period of 2 and an even function. In the interval f(x)=2x+ 1, find f(x) in the interval (? (2? x+4+ 1)(？ 2? x？ 0))