How to judge the parity of mathematical functions based on the x exponent? To explain in detail...

To judge the parity of a function, we first test whether the definition domain is symmetric 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.

If an odd function f(x) is meaningful at x=0, then the function must have a value of 0 at x=0. And about the origin symmetry.

If the function definition domain is not symmetrical with respect to the origin or does not meet the conditions of odd or even functions, it is called a non-odd and non-even function. For example, f(x)=x [-2] or [0,+ defines the domain is not symmetric with respect to the origin).

If a function conforms to both odd and even functions, it is called both odd and even. For example, f(x)=0

Note: Any constant function (which defines the domain symmetry with respect to the origin) is even, and only f(x)=0 is both odd and even.

Features. Overview.

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.

f(x) is the odd function "==" and the image of f(x) is symmetrical 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.

Even numbers exponential are even functions, and odd numbers exponential are odd functions.

To determine the parity of a function, you can judge it by the definition of the function or the characteristics of the image. Here are some commonly used methods:

1.Definition of odd and even functions: Judged by the definition and properties of the function.

A function f(x) is an odd function if and only if for any x, f(-x) =f(x) holds, i.e., the function is symmetric with respect to the y-axis. And a function f(x) is a grandchild puppet function, if and only if for any x, f(-x) =f(x) holds, i.e., the function is symmetric with respect to the origin.

2.Judgment of the image of a function: Observe the symmetry of a function on the image to determine its parity. For odd functions, the image is symmetrical with respect to the origin, i.e., left-right symmetry; For even functions, its image is symmetrical with respect to the y-axis, i.e., left-right symmetry.

3.Zero-point symmetry: For odd functions, if the function has a zero point x = a, then the corresponding function value also has a zero point of x = a; For even functions, if the function has a zero point x = a, then the corresponding function value also has a zero point of x = a.

4.For a derivative function, if the function f(x) is an even function, then its derivative f'(x) is an odd function; If the function f(x) is odd, then its derivative f'(x) is an even function.

It is important to note that these methods only work for functions that meet certain conditions, such as symmetry and derivability. For some complex functions, it may not be possible to judge directly from definitions and images, and it may be necessary to determine the parity of the functions through operations and complex analysis. In addition, for some functions, they are neither odd nor even, and their characteristics can be judged by specific calculations and property analysis.

Under similar conditions, the symmetry conclusion of the function f (Yuling roll x) can be obtained in a similar way.

For reference, please smile.

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Note: Parity only works on x

That is, when x becomes -x in the analytic formula, the corresponding function values are either equal or opposite to each other.

1. Decompose the function first into common general functions, such as polynomial x n, trigonometric functions, and determine parity.

2 Judging by the rules of operation between the decomposed functions, there are generally only three kinds of f(x)g(x), f(x)+g(x), and f(g(x)) (division or subtraction can be turned into corresponding multiplication and addition).

3 If one of f(x) and g(x) is an odd function and the other is an even function, then f(x)g(x) odd, f(x)+g(x) is non-odd and non-even, and f(g(x)) is odd.

4 If f(x) and g(x) are even functions, then f(x)g(x) even, f(x)+g(x)even, f(g(x)) even.

5 If f(x) and g(x) are odd functions, then f(x)g(x) even, f(x)+g(x)odd, f(g(x)) odd.

There are two ways to judge the parity of a function, one is to use a function image, if you can quickly draw a function image, as long as the image is symmetric about the y-axis, then it is an even function, and if the image is symmetrical about the origin, then it is an odd function. Another way to do this is to do it with a definition, which is a two-step process. The first step is to look at the definition domain, if the definition domain is about zero symmetry, then the next step, if the definition domain is asymmetric, it is a non-odd and non-even function.

The second step is to see f(-x)=f(x), which is an even function; If f(-x)=-f(x), it is an odd function.

The first root number in your question is x -2.

In this question, use definitions. Let's first look at the definition domain, x -2 0 and 2-x 0, and the solution is: the definition domain is {- 2, 2}, and there are only two elements.

About zero symmetry, of course. To do the second step, obviously f(-x)=f(x). So it's an even function.

Doesn't match the teacher's answer, unless you write the question incorrectly. Do it yourself in the right way, and believe in yourself.

The true exponential function y=a x is a non-odd and non-even function.

But y=a |x|is an even function.

When a function is symmetrical in its domain with respect to the origin, and there is f(-x)=f(x) in the defined domain, then it is an even function.

When a function has a domain that is symmetric about the origin, and there is f(-x)=-f(x) in the defined domain, then it is an odd function.

Exponential functions are non-odd and non-even functions.

The exponential function is one of the important fundamental elementary functions. In general, the function of y=a x (a is a constant and a>0,a≠1) is called an exponential function, and the domain of the function is r. Note that in the definition expression of the exponential function, the coefficient before a x must be the number 1, and the independent variable x must be in the position of the exponent, and cannot be any other expression of x, otherwise, it is not an exponential function.

The exponential function is defined in the domain r, provided that a is greater than 0 and not equal to 1. If a is not greater than 0, it will inevitably make the definition domain of the function discontinuous, so we do not consider it, and the function that a is equal to 0 is meaningless is generally not considered.

The exponential function has a range of (0, +, and the function graph is concave.

a>1, the exponential function increases monotonically; If 0 when a moves from 0 to infinity (not equal to 0), the curve of the function moves from a position close to the monotonically decreasing function of the positive half axis of the y axis and the positive half axis of the x axis to a position close to the position of the monotonically increasing function of the positive half axis of the y axis and the negative half axis of the x axis, respectively. where the horizontal line y=1 is a transition position from decreasing to increasing.

Functions always tend infinitely towards the x-axis in one direction and never intersect. The exponential function is unbounded. Exponential functions are non-odd and non-even functions. An exponential function has an inverse function, and its inverse function is a logarithmic function.

The above content refers to: Encyclopedia - Exponential Function.

Although the exponential function definition domain is symmetrical about the origin, the whole image is not symmetrical about the y-axis or the origin, so it does not have parity, but the quasi-exponential function is not necessarily, you can just bring a few values to look at it.