Does even function have an inverse function, and what is the relationship between a function and an

Because the even function must satisfy the requirement of f(-x) = f(x).

So for defining the domain.

When x=x0 and x=-x0, the function value is equal.

So when you find the inverse function, you will have an independent variable.

If there are two function values, the definition of the function does not match. So there is no inverse function for even functions.

However, it is possible to take some monotonic branch of the even function to find the inverse function.

For example, the function f(x)=x is an even function and has no inverse function. But if one takes a segment of this function f(x)=x (x 0) to find the inverse function, we get the inverse function g(x)= x(x 0) of f(x)=x(x 0). But it's clear that f(x)=x(x 0) is no longer an even function (the definition domain is not symmetrical with respect to the origin.

Definitely not. The even function corresponds to 2 different x's for a y, so his inverse function has 2 different y's for an x (the inverse function is to swap x, y), which violates the definition of a function, so there is none.

Yes, e.g. y=x 2 has an inverse function at (0, infinity).

This is not true, is the even function definition for a f(x) has f(x)=f(-x) f(-x) is not defined or is it an even function?

function withInverse functionsAbout symmetry with respect to y=x. If (a,b) is any point on the image of y=f(x), i.e. b=f(a). According to the definition of the inverse function, there is a=f-1(b), i.e., the point (b,a) on the image of the inverse function y=f-1(x).

The points (a,b) and (b,a) are symmetrical with respect to the straight line y=x, and from the arbitrariness of (a,b), we can know that f and f-1 are symmetrical with respect to y=x.

Quality. 1) The function f(x) is symmetrical with respect to the straight line y=x with respect to its inverse function f-1(x); The graph of the function and its inverse function is symmetrical with respect to the straight line y=x.

2) There are sufficient and necessary conditions for the function to have an inverse function.

Yes, the domain in which the function is defined.

It is a one-to-one mapping with the value range.

3) The tremor of one function and its inverse function are monotonous in the corresponding interval.

Unanimous. 4) Most of the even functions.

There is an inverse function (when the function y=f(x) and the domain is and f(x)=c (where c is a constant), then the function f(x) is an even function and has an inverse function, and the domain of the inverse function is , and the domain of the inverse function is , and the domain of the value is ). Odd functions.

There is no inverse function, and there is no inverse function when a straight line perpendicular to the y-axis can pass 2 or more points. If an odd function has an inverse function, its inverse function is also an odd function.

5) The monotonicity of a continuous function is consistent within the corresponding interval.

6) The function of strict increase (decrease) must have the inverse function of strict increase (decrease).

7) The inverse function is reciprocal and unique.

8) Define the law of opposite correspondence between domain and value range.

Reciprocal reversal (three reverses).

9) The derivative relation of the inverse function: if x=f(y) is in the open interval.

I is strictly monotonous, leadible, and f'(y) ≠0, then its inverse function y=f-1(x) is also derivative in the interval s=.

10) The inverse function of y=x is itself.

Not all of the Jan wheel functions have an inverse function. In the definition of a function, each value in the definition field of the letter can only correspond to the y value in the unique value range. So if the function has an inverse function, it is only possible if and only if for each y value in the value domain, it corresponds to the unique x value in the defined domain.

That is to say, different x's cannot be mirrored to shoot the same y function to have an inverse function.

The original function domain is the inverse function definition domain, and the original function definition domain is the inverse function domain, and they are also monotonicity in their respective definition domains. For a function, its inverse function is also a function, and according to the definition of an inverse function, it can be concluded that the original function is the inverse function of its inverse function, so for a function, the original function and the inverse function are called inverse functions to each other.

There is no backcopy function for even functions.

Because the even function must satisfy the requirement of f(-x) = f(x).

So for a non-0 x0 in the defined domain, when x=x0 and x=-x0, the function value is equal.

Therefore, when finding the inverse function, there will be a situation where one independent variable corresponds to two function values, which does not meet the definition of "one-to-one correspondence within the definition of the inverse function". So there is no inverse function for even functions.

However, it is possible to take some monotonic branch of the even function to find the inverse function.

For example, the function f(x)=x is an even function and has no inverse function. But if one takes a segment of this function f(x)=x (x 0) to find the inverse function, we get the inverse function g(x)= x(x 0) of f(x)=x(x 0). But it's clear that f(x)=x(x 0) is no longer an even function (the definition domain is not symmetrical with respect to the origin.

Says who? The inverse function of the cosine function is the inverse cosine function.

If the function is in a monotonic interval, there is an inverse function.

To solve such a problem, we need to grasp the definition

The inverse function is defined as follows: in general, if x corresponds to y with respect to some correspondence f(x), y=f(x), then the inverse function of y=f(x) is y= f '(x).

The condition for the existence of an inverse function is that the original function must be "one-to-one", like a target: a person has only one bullet, and it corresponds to a target; Someone hits every target, and only one person hits the "inverse function" It's like hitting people in reverse, pulling away ......

If there is no one-to-one correspondence, the "reverse" may occur later: one number, two numbers corresponding to it

Also, it is important to know that the image of the inverse function and the original function is symmetrical with respect to y=x

Look at (1): The odd function is symmetrical with respect to the origin, and it is a one-to-one correspondence, considering that it is folded with respect to y=x, it is obviously still symmetrical with respect to the origin

The even function is not good, because it is not a "one-to-one correspondence" When x takes x0 and -x0, it corresponds to the same y0, and you are the opposite, and he y0 does not know who it corresponds to (it must be unique, otherwise it is not a function).

Therefore, if the original function is an odd function, so is the inverse function; The original function is an even function, and there is no inverse function

Look at (2) again:

Obviously not, if you look at the definition of parity, there is only an ethereal "f(x)" in the definition, and there is no requirement for anything, so there is no need to be monotonous

1. It is not the same, even functions do not have a global inverse function.

For example, y=x, there are two inverse functions, x= y or x=- y, both of which have no parity properties.

2. y=sinx, periodic non-monotonic function. Odd functions.

One counterexample is sufficient.