What are the knowledge points of the common test of mathematical function parity in the first year o

1 Definition of Functions.

In general, for the function f(x).

1) If there is f(-x) = f(x) for any x in the function definition domain, then the function f(x) is called an odd function.

2) If there is f(-x)=f(x) for any x in the function definition field, then the function f(x) is called an even function.

3) If f(-x)=-f(x) and f(-x)=f(x) are true at the same time as f(-x)=f(x) for any x in the function definition domain, then the function f(x) is both odd and even, and is called both odd and even.

4) If f(-x)=-f(x) and f(-x)=f(x) cannot be true for any x in the function definition domain, then the function f(x) is neither odd nor even, and is called a non-odd and non-even function.

Note: Odd and evenness are integral properties of a function, for the entire defined domain.

The domain of an odd and even function must be symmetrical with respect to the origin, and if the domain of a function is not with respect to the symmetry of the origin, then the function must not be an odd (or even) function.

Analysis: To judge the parity of a function, first test whether the definition domain is symmetrical with respect to the origin, and then simplify and sort it out in strict accordance with the definition of odd and evenness, and then compare it with f(x) to draw conclusions).

The basis for judging or proving whether a function is parity is by definition.

2 Characteristics of parity images.

Theorem singular function image with respect to the origin of the center symmetric graph, even function image with respect to the y axis or axisymmetric graph.

f(x) is the odd function "f(x)'s image symmetry with respect to the origin.

Point (x,y) (x,-y).

If the odd function increases monotonically over an interval, it also increases monotonically on its symmetrical interval.

Even functions that increase monotonically over a certain interval decrease monotonically in its symmetrical interval.

3.Parity function operations.

1).The sum of two even functions is an even function.

2).The sum of two odd functions is an odd function.

3).The sum of an even function and an odd function is a non-odd function and a non-even function.

4).The product of two even functions multiplied is an even function.

5).The product of two odd functions multiplied is an even function.

For example, you are given a function to determine whether the function refers to the outburst as an odd or even function; Or give Lu Xiang a function equation with unknown parameters, and then give you the parity of this function, as well as some other false hints to let you find the unknown.

Definition of parity of functions: For the function elimination of the prudent number f(x), if for any x in the field of the function is f(-x)=-f(x) (or f(-x)=f(x)), then the function f(x) is called an odd function (or even function).It is important to understand the definitions of odd functions and even talk functions correctly.

The definition of parity function is the main basis for judging the parity of a function. In order to determine the parity of a function, it is sometimes necessary to simplify the function or apply the equivalent form of the definition: Note the application of the following conclusions.

Some properties and conclusions about parity (1) The sufficient and necessary condition for a function to be an odd function is that its image is symmetrical with respect to the origin.

；f(x) x to the power of -4 +2; f(x)＝3;f(x)=2x to the power of 4 + 3x is an even function.

f(x) x-1 2 non-odd non-even Define domain asymmetry 2...f(x)=x to the 3rd power; f(x)=x to the power of 5—2x is an odd function; f(x)=x to the power of 3 + x; f(x)=x +1(x [ 1,3]) is not odd or even.

f（x）=|x+2|-|x-2|Odd functions.

Prerequisites define domain symmetry.

f(x)=f(-x) is an even function.

f(x)=-f(-x) is an odd function.

Note that the proportional function is odd.

Proportional function Odd function.

Inverse proportional function Odd function.

Sinusoidal function Odd function.

Cosine function Even function.

A function b that is not 0 is not odd or even.

Power functions are possible in all three types of exponents are even, even positively odd, odd and negative, only in the first quadrant there is an image, non-odd and non-even.

Exponential functions, non-odd and non-even.

Tangent function, odd function.

First, determine whether the defined domain is symmetrical with respect to the origin, and asymmetry is a non-odd and non-even function.

When it is done, it is judged that f(-x)=f(x) is an even function, and f(-x)=-f(x) is an odd function, and both are in line with both odd and even functions.

If the domain is defined with respect to the origin symmetry, but does not conform to the above formula, it is also a non-odd and non-even function.

Define generally, for the function f(x).

1) If there is f(-x)=f(x) for any x in the function definition field, then the function f(x) is called an even function. With respect to y-axis symmetry, f(-x) = f(x).

2) If there is f(-x)=-f(x) for any x in the function definition domain, then the function f(x) is called an odd function. With respect to origin symmetry, -f(x) = f(-x).

3) If for any x in the function definition domain, there are f(-x)=-f(x) and f(-x)=f(x), (x d, and d is symmetrical with respect to the origin. Then the function f(x) is both odd and even, and is called both odd and even.

4) If f(-x)=-f(x) and f(-x)=f(x) cannot be true for any x in the function definition domain, then the function f(x) is neither odd nor even, and is called a non-odd and non-even function.

x is replaced by -x, the same value is even, the opposite number is odd, the first two premises are the definition of the domain symmetry with respect to x=0, otherwise it is non-odd and non-even.

x<0 then -x>0

So f(-x) applies to the equation x>0.

So f(-x) = -x +2x -1

Odd function f(x) = -f(-x).

and the odd function f(0)=0

So f(x)=

x³-2x²+1，x<0

0，x=0x³+2x²-1，x>0

It is known that f(x) and g(x)= are odd and even functions on (-a, a), respectively, so f(x)=-f(-x), g(x)=g(-x)

m(x)=f（x）·g(x)=-f(-x)*g(-x)=m(-x)

Let t(x)=f(x)·g(x) because f(x) and g(x)= are odd and even functions on (-a, a), respectively.

So f(-x)=-f(x) g(x)=g(x)so t(x)=f(x)·g(x).

t(-x)=f(-x)·g(-x)=-f(x)·g(x)so t(x)=-t(-x).

So it's an odd function.

Let f(x) be an odd function, g(x) be an even function, h(x)=f(x)·g(x), then f(x)=-f(-x), g(x)=g(-x), h(x)=f(x)·g(x)=-f(-x)·g(-x)=-h(-x).

h(x)=f(x)·g(x) is an odd function over (-a,a).

(1) Prov: first, f(1)=0; Needless to say, this goes without saying.

f(2*2)=f(-2*-2)=2f(2)=2=2f(-2);

f(-2)=1;then f(-2*-1)=f(-2)+f(-1)=1;

f(-1)=0;

f(-x)=f(-1)+f(x)=f(x);

Completion of the certificate; (2) Proof:

Let n>1; Rule.

f(x*n)=f(x)+f(n)

f(x*n)-f(x)=f(n);

n>1

f(n)>0;x*n>x(true when x>0);

f(x*n)-f(x)>0

This proves that when x>0, the function f(x) is constant; Thank you!