How can you tell if a function is parity? How to judge the parity of a function

In particular, the parity of a function is only for a function alone, and the function in this problem.

y=log3^x

y=3 x are two functions in their defined domain, which can only be said to be symmetrical with respect to the straight line y=x, and cannot be said to be parity. Neither of these functions is odd nor even.

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) or 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 of all, to test whether the definition domain is symmetrical with respect to the origin, and then to 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) References:

First, according to the image, if the image is symmetric with respect to the y-axis, it is an even function,。。 If it is symmetrical about the origin, it is an odd function...

You can also use algebra ... According to f(x)=f(-x), we can see that this is an even function.

According to f(-x)=-f(x), it can be seen that this is an odd function.

In the case where the domain is r, f(0)=0 can also be used to determine that it is an odd function.

The parity method of judging a function is described as follows:

1. According to the odd function.

and the definition of even functions.

f(-x) = f(x), it is an even function; f(-x) = f(x), it is an odd function.

2. Judge according to the image of the function.

The image of the function is symmetrical with respect to the y-axis (the domain of the definition of the function.

must be symmetrical about the origin), then it is an even function; The image of the function is symmetrical with respect to the origin center (the domain of the function must be symmetrical with respect to the origin), then it is an odd function.

Monotonicity of parity functions over symmetric intervals.

Characteristics of the value range. 1. The monotonicity of odd functions in the symmetry interval is the same, and the monotonicity of even functions in the symmetry interval is opposite.

2. The value range of the odd function on the symmetry interval is symmetrical with respect to the origin, and the value range of the even function on the symmetry interval is the same.

In particular, if an odd function has 0 in its definition domain, then there must be f(0)=0.

1.According to the odd function.

and the definition of even functions.

f(-x) = f(x), it is an even function;

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

2.Judgment is made based on the image of the function.

The image of the function is symmetrical with respect to the y-axis (the domain of the function must be symmetrical with respect to the origin), then it is an even function;

The image of the function is symmetrical with respect to the origin center.

The domain of the function must be symmetric about the origin), then it is an odd function.

1. Image judgment, the odd function image is symmetrical with respect to the origin, and the even function image is symmetrical with respect to the y-axis.

2. Define the judgment, the odd function f(-x) = -f(x).

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

Odd function f(-x) = -f(x) even function f(x) = f(-x) question. <>

The fourth and fifth sub-questions.

Questions. I don't understand where this idea came from.

Put this formula into the odd function f(-x) = -f(x) and the even function f(x) = f(-x), if you have two formulas like the fourth problem, put it in twice.

First look at whether the defined domain is symmetrical with respect to the origin, otherwise it is a non-product non-even function, and then calculate the relationship between f(-x) and f(x), if the even function: f(-x)=f(x).

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

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

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

Theorem The image of the odd function is a symmetric graph with respect to the origin, and the image of the even function is axisymmetric with respect to the y-axis.

For the function f(x) about x, regardless of its analytic formula.

If the function f(-x) obtained by substituting -x for x satisfies :

f(-x) f(x), then the function is even.

f(-x) -f(x), then the function is odd.

Of course, the function f(x) has an important restriction, which is that the lower and upper limits of the domain should be inverse to each other (format: (-a,a),a 0). If none of its defined domains satisfy this condition, it can even be concluded that f(x) is neither odd nor even without the above judgment.

First of all, we must first find the definition domain of the function, the definition domain must be symmetric with respect to the origin, and then we should see that when the independent variable takes the opposite number, the corresponding function value is odd on the opposite side, and equal is even.

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

By definition, the parity of a function can be determined.

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

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

Parity is judged by the original function.

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

Odd function: f(-x)=-f(x), continuous odd function must pass the origin.

Parity is judged as follows:

1. Definition method.

Use definitions to judge function parity.

is the main method, which first finds the definition domain of the function.

Observe and verify whether there is symmetry about the origin. Secondly, the function 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.

A defined domain with parity must be symmetrical with respect to the original point, 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.

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

Degree. If the image of f(x) is symmetric with respect to the y-axis, then f(x) is an even function.

4. Operate with functions.

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".

Even functions on symmetric intervalsMonotonniaIt's the opposite.

The monotonicity of odd functions is consistent across the entire defined domain. The sum of two even functions is an even function, and the sum of two odd functions is an odd function.

The product of two even functions multiplied is an even function, the product of two odd functions multiplied is an even function, and the product of an even function multiplied by an odd function is an odd function.

Several functions are compounded, as long as one of them is an even function, the result is an even function; If there is no even function, it is an odd function, and the sum and difference product quotient of an even function is an even function.

The sum difference of an odd function is an odd function, the even-numbered product quotient of an odd function is an even function, the odd product quotient of an odd function is an odd function, and the absolute value of an odd function.

is an even function, and the absolute value of the even function is an even function.

Solution:

y=x^3。

Range

y (negative infinity, positive infinity).

Parity

y=x 3 is an odd function.

Monotonnia

y=x 3 increases monotonically.

Tips for learning math.

1. Be good at thinking when learning mathematics, and the answers you come up with are far more impressive than the answers told by others.

2. Do a good job of preview before class, so that you can better digest and absorb the knowledge points when you take math class.

3. Mathematical formulas must be memorized, and they must be able to derive and draw inferences.

4. The most basic thing to learn mathematics well is to master the knowledge points of the textbook and the exercises after class.

5. 80% of the scores in mathematics are in the basic knowledge, and 20% of the scores are difficult, so it is not difficult to score 120 points.

To determine the parity of a function, you can take the following approach:

1.Using the phonetic search definition of the function, it is judged that a function f(x) is an odd function if and only if f(-x) =f(x) is true for all x.

In other words, if the independent variable of a function is taken as the opposite number, and then the value of the function is also taken as the opposite number, then the high-ranking friend function is an odd function.

2.Judge by using the function image: If a function is symmetrical with respect to the origin, i.e. the image is symmetrical with respect to the origin, then the function is an odd function.

In other words, if you flip the function image 180 degrees along the y-axis, then the image does not change, then the function is odd.

3.Use function expressions to judge: The parity of some functions can be inferred directly from their function expressions. For example, a polynomial function that contains only terms to the odd power is an odd function, and a function that contains only an even power term is an even function.

It is important to note that some functions are neither odd nor even, and such functions are called general functions. In addition, some functions are odd or even in a specific interval, while parity is not satisfied in other regions. Therefore, when judging the parity of a function, it is necessary to consider the information such as function definition, image, and expression.

Judging the parity of a function helps to simplify the analysis and solution of the function. In practical problems, parity can also be used to simplify calculations and obtain some important functional properties.

In the study of functions, we often discuss their symmetry. Symmetry can help us understand the nature and characteristics of the function image. Here are five common functional symmetry conclusions and their derivations:

1.Even Functions:

If a function satisfies f(x) = f(-x) for any x, i.e., symmetry with respect to the y-axis, then the function is called an even letter circular search.

2.Odd Functions:

If a function satisfies f(x) = f(-x) for any x, i.e., symmetry with respect to the origin, then the function is called an odd function.

3.Periodic Function:

If a function satisfies f(x + t) = f(x) for a certain constant t and all x, then the orange calendar is called a periodic function. t is known as the period of the function.

4.Axis of Symmetry:

If a function has an axis of symmetry, i.e., there is some real number a, and when x=a, the image of the function is symmetrical with respect to the axis of symmetry, then the function has an axis of symmetry.

5.Center Symmetry:

If a function satisfies f(a + x) =f(a - x) for some real number a and all x, i.e., with respect to the line x=a symmetry, then the function is said to be centrally symmetrical.

These five conclusions can be deduced from images, changes in functional relationships, or definitions. By observing and analyzing the properties of functions, it is possible to determine whether a function has symmetry and the specific type of symmetry. The derivation of the symmetry conclusion helps us to better understand and study the characteristics of the delay function and its image.