How to distinguish between odd and even functions, and how to judge odd and even functions

The odd function has the same monotonicity on its symmetric intervals [a,b] and [-b,-a], i.e., if it is known to be an odd function, it is an increasing function (subtracting function) on the interval [a,b], and it is also an increasing function (subtracting function) on the interval [-b,-a];

Even functions have opposite monotonicity in their symmetric intervals [a,b] and [-b,-a], i.e., if they are known to be even functions and are increasing (subtracting) on intervals [a,b], they are subtractive (decreasing functions) on intervals [-b,-a]. But monotonicity cannot be reversed from parity. Verification of parity requires that the domain of the function must be symmetric with respect to the origin.

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 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.

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

The odd function is symmetrical with respect to the origin and the even function is symmetric with respect to the y-axis.

There is also the formula odd function f(x) = f(-x) even function f(x) = -f(x) Replace x with a number and bring it in and try these two formulas.

The odd function refers to the fact that for any x in the definition domain of the function f(x) about the origin symmetry, there is f(-x) = - f(x), then the function f(x) is called odd function.

In general, if there is f(x)=f(-x) for any x in the definition domain of the function f(x), then the function f(x) is called even function.

Nature 1The difference between the sum or subtraction of two odd functions is the odd function.

2.The difference between the sum or subtraction of an even function and an odd function is a non-odd and non-even function.

3.The product of two odd functions multiplied or the quotient obtained by division is an even function.

4.The product of an even function multiplied by an odd function or the quotient obtained by division is an odd function.

Algorithms. 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.

6) The product of multiplying an even function by an odd function is an odd function.

Odd Functions:The odd function refers to the fact that for any x in the definition domain of the function f(x) about the origin symmetry, there is f(-x) = - f(x), then the function f(x) is called odd function.

Even Functions:In general, if there is f(x)=f(-x) for any x in the definition domain of the function f(x), then the function f(x) is called even function.

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.

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

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

1.The domain of a function is defined as an interval about origin symmetry, e.g. (a,a),(a,a].

If the defined domain is not symmetrical with respect to the origin, it is a non-odd and non-even function).

2.For any number x in the defined domain, if f(-x) f(x), the function f(x) is said to be odd on the defined domain;

If f(-x) f(x), the function f(x) is said to be even in the defined domain (the sign of the three horizontals represents an constant equal).

3.Graphically speaking, the graph of the odd function is symmetrical with respect to the origin, and the graph of the even function is symmetric with respect to the y-axis.

First, define the domain to be symmetrical.

Denial is neither an odd nor an even function.

After that, let's look at what symmetry is.

With respect to origin symmetry is the odd function.

With respect to y-axis symmetry is an even function.

Odd and even functions are judged as follows

1. From the definition point of view:

In general, if there is f(-x)=f(x) for any x in the definition domain of the function f(x), then the function f(x) is called an even function.

In general, if there is f(-x)=-f(x) for any x in the domain where the function f(x) is defined, then the function f(x) is called an odd function.

2. From the image:

The tuxiang of the even function is symmetric with respect to the y-axis, and the graph of the odd function is symmetrical with respect to the origin.

f(x) is the odd function "The image of f(x) is about the origin symmetry point (x,y) (x,-y) The odd function increases monotonically in a certain interval, and it also increases monotonically in its symmetry interval.

Image characteristics of odd and even functions

1. The odd function image is symmetrical with respect to the origin. The image of the odd function is a central symmetrical image with the origin as the center of symmetry.

2. The image of the even function is symmetrical with respect to the y-axis. The image of an even function is an axisymmetric image with the y-axis as the axis of symmetry.

3. The monotonicity of odd functions in the symmetry interval is the same, and the monotonicity of even functions in the symmetry interval is opposite.

If the domain of a function f(x) is d, and for any x d, there is -x d, and f(-x)=-f(x), then f(x) is said to be an odd function. If f(-x) = f(x), then f(x) is an even function.

OddEven Functions:The method of judging is as follows:

1. Definition method judgment. Use definitions to judge function parity.

is the main method. First, find the defined domain of the function.

Observe and verify whether there is symmetry about the origin. Secondly, the function formula is simplified, then f(-x) is calculated, and finally the parity of f(x) is determined according to the relationship between f(-x) and f(x).

2. Use the necessary conditions.

Judgment. A defined domain with parity must be symmetric with respect to the origin, which is a necessary condition for the function to have parity. For example, the definition domain of the function y= (-1) (1, + the definition domain is asymmetrical with respect to the origin, so this function is not parity.

3. Use symmetry to judge. If the image of f(x) is symmetrical with respect to the origin, then f(x) is an odd function.

If the image of f(x) is symmetrical with respect to the y-axis.

If this drawback, f(x) is an even function.

4. Use function operation to judge. If f(x), g(x) are odd functions defined on d, then on d, f(x)+g(x) are odd functions, and f(x) g(x) are even functions. Simply, "odd + odd = odd, odd = even".

Similarly, "even = even, even = even, even = even, odd = odd".