First, the sort of knowledge

1, the nature of the function: domain and range [extremum, most value], monotonicity, parity, periodicity, symmetry, zero;

2, basic elementary functions: constant function, a power function, an exponential function, logarithmic functions, trigonometric functions;

3, various properties are given by:

Common monotonicity given by way

1, given in the form of an image;
2, the subject is given directly by the literal language;
3, to define a given formula;
4, in an equivalent modification of formula defined in the form of the product [] is given;
5, in the form of defined deformation equivalents [] given by the quotient;
6, the conclusions given in the form of monotonic function;
7, given in the form of derivative,

Given by way of common parity

1, given directly;
2, to define a given formula;
3, the definition given in the form of variable;
4, in the form of [] function or segmented form of an image is given;
5, the conclusions is given in the form of parity nature of the application;
6, in the form of a portion integral with the given parity,
7, the image is converted to the basis is given,

Common odd function:

$f(x)=kx$；

$f(x)=x^3$；

$ F (x) = x ^ k (k is odd) $;

Y = Asin $ \ omega x $;

$ Y = e ^ x ^ {- x} $;

$y=2^x-2^{-x}$；

$y=ln\frac{x+1}{x-1}$；

$f(x)=x+\frac{k}{x}(k\neq 0)$；

$g(x)=lg(\sqrt{sin^2x+1}+sinx)$；

$g(x)=x^3+lg(\sqrt{x^2+1}+x)$；

$f(x)=x^3\pm 3sinx$

$f(x)=ln(\sqrt{x^2+1}-x)$；

Common dual function:

$f(x)=x^2$；

$y=k|x|(k\in R)$；

$ Y = e ^ {| x |} $;

$ F (x) = x ^ k (k is an even number) $;

$ Y = Acos \ omega x + k $;

$ Y = e ^ x + e ^ {- x} $;

$y=2^x+2^{-x}$；

$ F (x) = LN (1+ | x |) $;

$f(x)=\frac{|x|}{x^2+1}$

Given by way of conventional cyclical

1, given in the form of an image;
2, to define a given period of formula;
3, the conclusions given periodically;

Given by way of common symmetry

1, given in the form of an image;
2, given in the form of parity [parity is a special case of the symmetry];
3, given in the form of expansion of parity;
4, given in the form of periodic + parity;

Clearance cognitive, distinguished in three kinds of confusion

[Integral] periodic addition of two independent variables can not eliminate the \ (X \) on the performance of periodic;

As described by \ (F (X + 2) = F (X) \) , then \ (T = 2 \) , as indicated by \ (F (X + 2) = - F (X) \) , then \ (T = 4 \) ,

[] Symmetry of the overall sum of the two arguments can eliminate \ (x \) on the performance of symmetry;

As described by \ (F (the -X-) + F (X) = 0 \) , the center of symmetry \ ((0,0) \) , i.e. the odd function; special symmetry.

As described by \ (f (4-x) + f (x) = 2 \) , for the center of symmetry \ ((2,1) \) , i.e., the general symmetry, the center of symmetry;

As described by \ (F (the -X-) -f (X) = 0 \) , the axis of symmetry \ (X = 0 \) , i.e. dual function, special symmetry;

As described by \ (F (X-2) -f (X) = 0 \) , the axis of symmetry \ (X =. 1 \) , i.e., the general symmetry axis of symmetry;

Thinking blind spots

Parity function, symmetry, three periodic nature, wherein two long known, the third can be deduced, and the third is often essential in solving problems, so we need to open up the thinking blind spots, distortion and proficiency in the following mathematical thinking:

Symmetry parity + \ (\ Longrightarrow \) periodical modification of

For example, a known function \ (f (x) \) is an odd function, satisfying \ (F (X-2) = F (X) \) ,

则由$\begin{align*} f(2-x)&=f(x) \\ - f(-x)&= f(x)\end{align*}$ $\Bigg\}\Longrightarrow f(2-x)=- f(-x)\Longrightarrow f(2+x)=- f(x)\Longrightarrow$周期$T=4$

Periodically parity + \ (\ Longrightarrow \) modification of symmetry

For example, a known function \ (f (x) \) is an odd function, satisfying \ (F (X +. 4) = - F (X) \) ,

By \ (\ align = left the begin {F} * (X +. 4) & = - F (X) F \\ (the -X-) & = - F (X) \ * End align = left {} \) \ (\ Bigg \ } \ Longrightarrow f (x + 4 ) = f (-x) \ Longrightarrow \) symmetry axis is \ (x = 2 \)

+ Periodic symmetry \ (\ Longrightarrow \) modification of parity

For example, a known function \ (f (x) \) period is 2, and satisfies \ (F (X + 2) = F (the -X-) \) ,

By \ (\ align = left the begin {F} * (2 + X) = & F (the -X-) \\ F (X + 2) & = F (X) \ * End align = left {} \) \ (\ Bigg \ } \ Longrightarrow f (-x) = f (x) \ Longrightarrow \) function \ (f (x) \) is an even function.

Second, the examples election talk

Example 1 [2016 Matriculation Mathematics 2 Paper for Title] [12] a common axis of symmetry of known function \ (f (x) (x \ in R) \) satisfies \ (F (X) = \) \ (F (2- X) \) , if the function \ (y = | x ^ 2-2x -3 | \) and the function \ (y = f (x) \) the intersection of the image is \ ((x_1, Y_1) \) , \ ( (x_2, Y_2) \) , \ (\ cdots \) , \ ((x_m, Y_M) \) , then \ (\ sum \ limits_ {i = 1} ^ m {x_i} \) values []

$A.0$ $B.m$ $C.2m$ $D.4m$

Example 2 [2017 Arts 1 Paper for Item 9] entrance Zhenti known function \ (F (X) = LNX + LN (X-2) \) , then []

$ A. $ in $ (0,2) $ on monotonically increasing

In B. $ $ $ (0,2) $ monotonic decreasing

$ Image Cy = f (x) $ about $ x = 1 $ linearly symmetrical

Image $ Dy = f (x) $ $ about point (1,0) $ symmetry

Example 3 [2017] [Baoji function of two properties of the subject is given one by one is known] is defined in \ (R & lt \) function on a \ (y = f (x) \) satisfies the following condition:

① for any \ (X \ in R & lt \) , are \ (F (X + 2) = F (X-2) \) ;

② function \ (y = f (x + 2) \) is an even function;

③ When \ (x \ in (0,2] \) when, \ (F (X) = X-E ^ \ cfrac. 1} {X} {\) ,

If the known \ (A = F (-5) \) , \ (F B = (\ {cfrac. 19} {2}) \) , \ (C = F (\ cfrac 41 is {{}}. 4) \) , the \ (a \) , \ (B \) , \ (C \) size relationship []

$A.b < a < c$ $B.c < a < b$ $C.c < b < a$ $D.a < b < c$

Example 4 [2019] Ning analog known function \ (f (x) \) is the domain of \ (R & lt \) , and in \ ([0, + \ infty ) \) monotonically increasing on, \ (G (X) -f = (| X |) \) , if the \ (G (LGX)> G (. 1) \) , then \ (X \) is in the range []

$A.(0,10)$ $B.(10,+\infty)$ $C.(\cfrac{1}{10},10)$ $D.(0,\cfrac{1}{10})\cup (10,+\infty)$

[Description] See the rest of the weekend to explain the subject of regular training 03.

Integrated application function of the nature of [the weekend Lecture Outline]