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Anonymous users2024-02-06If 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 odd function.
Odd function properties:
1. The image is symmetrical with respect to the origin.
2. F(-x) is satisfied
f(x)3, monotonicity is consistent in the interval with respect to the origin symmetry.
4. If the odd function is defined in x=0, then there is f(0)=05, and the domain is symmetrical with respect to the origin (common to odd-even functions).
If you know the function expression, f(x)=f(-x) is satisfied as y=x*x, y=cosx, if you know its function image, the even function image is symmetrical with respect to the y-axis (x=0).
The domain of an even function must be symmetric with respect to the origin, otherwise it cannot be an even function.
Even function properties:
1. The image is symmetrical with respect to the y-axis.
2. F(-x) is satisfied
f(x)3, the monotonicity is opposite in the interval with respect to the origin symmetry.
4. If a function is both odd and even, then there is f(x)=05, and the domain is symmetrical with respect to the origin (common to odd-even functions).
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Anonymous users2024-02-05Odd function properties:
1. The image is symmetrical with respect to the origin.
2. F(-x) is satisfied
f(x)3, monotonicity is consistent in the interval with respect to the origin symmetry.
4. If the odd function is defined in x=0, then there is f(0)=05, and the domain is symmetrical with respect to the origin (common to odd-even functions).
Even function properties:
1. The image is symmetrical with respect to the y-axis.
2. F(-x) is satisfied
f(x)3, the monotonicity is opposite in the interval with respect to the origin symmetry.
4. If a function is both odd and even, then there is f(x)=05, and the domain is symmetrical with respect to the origin (common to odd-even functions).
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Anonymous users2024-02-04A function that is both odd and even is a constant function that defines the domain with respect to the origin symmetry.
The function with respect to origin symmetry is an odd function, the function with respect to y-axis symmetry is an even function, and the sum obtained by adding two even functions is an even function. The sum of an even function and an odd function is a non-odd function and a non-even function. The product of two odd functions multiplied is an even function.
Odd function properties:
1. The 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 obtained by multiplying two odd functions 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.
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Anonymous users2024-02-03f(x)=c (c is a constant), when c ≠ 0, f(x) is only an even function, not an odd function. f(x) only satisfies the requirement of f(-x)=f(x), not f(-x)=-f(x).
Therefore, there is only one type of function that is both odd and even, that is, f(x)=0, and the domain is symmetric with respect to the origin, this kind of function satisfies both the requirements of f(-x)=f(x) and f(-x)=-f(x). So it's both an odd function and an even function.
Proof method: Since f(x) is both an odd and an even function, the domain is defined to be symmetric with respect to the origin.
When x=0, if f(x) is defined, because f(x) is an odd function, i.e., f(0)=-f(-0) holds, i.e., f(0)=-f(0) holds, and f(0)=0 is obtained.
When x≠0, since f(x) is an odd function, f(x)=-f(-x) holds; Because f(x) is also an even function, f(x) = f(-x).
So f(x)=-f(-x) and f(x)=f(-x) are both true, so we get f(x)=-f(x), so f(x)=0.
So f(x) is the constant equals 0 and defines the domain as a function of the origin symmetry.
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Anonymous users2024-02-02All constant functions of the domain with respect to the origin symmetry f(x)=0 are both odd and even.
If the function is both odd and even, then the function must be a constant function that defines the symmetry of the domain with respect to the origin f(x)=0.
i.e.: f(x)=0,x [-a,a] or x (-a,a) where a is any real number.
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Anonymous users2024-02-01f(x) is both an odd and even function, and the sufficient and necessary condition is that f(x)=0f(x) is both an odd and even function.
f(x)=f(-x)=-f(x).
f(x)=0
As long as the domain is symmetrical with respect to the origin, the corresponding rule is f(x)=0, which is both odd and even functions.
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Anonymous users2024-01-31The definite integral of an odd function on a symmetric interval is zero-even and the definite integral of a symmetric interval is twice that of half of its interval. This property is abbreviated as even odd zero.
Odd function properties:
1. The image is symmetrical with respect to the origin.
2. F(-x) = f(x).
3. The monotonicity is consistent in the interval of origin symmetry.
4. If the odd function is defined in x=0, then there is f(0)=05, and the domain is symmetrical with respect to the origin (common to odd-even functions).
Even function properties:
1. The image is symmetrical with respect to the y-axis.
2. F(-x) = f(x).
3. The monotonicity of the interval about the origin symmetry is reversed.
4. If a function is both an odd function and an even function, then there is f(x)=05, and the domain is symmetrical with respect to the origin (the odd and even functions are commemorated and dismantled) <>
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Anonymous users2024-01-30For example, if any x in the domain of the function f(x) has f(-x) = f(x), then the function f(x) is called an odd function. In general, if there is f(x)=f(-x) for an x of any bright rock stove in the defined domain of the function f(x), then the function f(x) is called an even function.
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Anonymous users2024-01-29Hunger in the grip. 1. Properties of odd functions:
1. The image is symmetrical with respect to the origin.
2. The monotonicity is consistent in the interval of origin symmetry;
3. Define the common properties of origin symmetry and odd-even functions in the rotten Qing domain.
2. Properties of even functions:
1. The image is symmetrical with respect to the y-axis.
2. The monotonicity is opposite in the interval of the origin symmetry;
3. The definite domain is symmetrical about the origin, and the common property of odd-even functions.
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Anonymous users2024-01-28The odd function is symmetrical in the heart of the Imperial Palace.
The even function is symmetrical left and right.
All properties are derived from this.
There are many odd function properties:
1. The image is symmetrical with respect to the origin.
2. F(-x) = f(x).
3. The monotonicity is consistent in the interval of origin symmetry.
4. If the odd function is defined in x=0, then there is f(0)=05, and the even function property of the definition domain is symmetrical with respect to the origin (the odd-even function is called together).
1. The image is symmetrical with respect to the y-axis.
2. F(-x) = f(x).
3. The monotonicity of the interval about the origin symmetry is reversed.
4. If a function is both odd and has a parallel field is an even function, then there is f(x)=05, and the domain is symmetric with respect to the origin (common to odd-even functions).
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Anonymous users2024-01-27Odd Functions Odd functions are even functions. The odd function multiplied by the odd function equals the even function. Odd function multiplication and even function are odd functions, odd function addition and subtraction of odd functions are odd functions, even function addition and subtraction of even functions are even functions, odd functions multiply odd functions are even functions, and even functions multiply even functions are even functions.
An even function multiplied by an even function is an even function.
1. Addition rules for parity functions1) The odd function plus the odd function with the orange number is the odd function.
2) The function obtained by adding the even function to the even function is the even function.
3) The function obtained by adding the odd function to the even function is the non-odd and non-even function.
2. Subtraction rules for odd-even functions1) The odd function is obtained by subtracting the odd function to obtain the odd function.
2) The even function is obtained by subtracting the even function.
3) The odd function minus the even function is a non-odd and non-even function.
3. Multiplication rules for odd and even functions1) The odd function multiplied by the odd function is an even function.
2) The odd function multiplied by the even function is the odd function.
3) The even function is multiplied by the even function to obtain the even function.
4. The division rules of odd and even functions1) The odd function divided by the odd function is an even function.
2) The odd function is divided by the even function to obtain the odd function.
3) The even function is obtained by dividing the even function.
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