Isn t it true that all functions have inverse functions

In the definition of a function, for Define Domain.

Each value can only correspond to the y value in a unique 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, different x's cannot be mapped to the same y function to have an inverse function.

The correct proposition is:

Reason. Because of x r, f(x+3 2)-f(x)=0 gives f(x)=f(x+3 2).

Then f(x) coincides with the original image every time it is translated by 3 2 units, so the period of f(x) is 3 2;

From the known y=f(x-3 4) as an odd function, f(-x-3 4)=-f(x-3 4)....①

Let t=x-3 4,t r

i.e. x=t+3 4

Substituting f(-t-3 2) = -f(t).

And the period of f(x) is 3 2, so f(-t)=-f(t), so f(t) is an odd function, symmetrical with respect to (0,0).

So the function y=f(x) of the image is symmetrical with respect to the point (3, 4, 0).

And the period of f(x) is 3 2, so the image of the function y=f(x) is symmetrical with respect to the point p(-3 4,0).

It is not possible to determine whether it is symmetrical (even function) on the y-axis. Therefore, it is considered incorrect.

For example, y=f(x)=tan(2 x 3) is an odd function.

y=f(x)=cos(4 x 3) is an even function.

They also meet the stem conditions.

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.

The inverse function is:Let the domain of the function y=f(x) be d and the range of values be f(d). If, for every y in the range f(d), there is only one x in d such that g(y)=x, then a function defined on f(d) is obtained according to this correspondence rule, and this function is called the inverse of the function y=f(x).

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 x=f-1(y). The condition for the existence of an inverse function (a single-valued function by default) is that the original function must be one-to-one (not necessarily within the entire number field). Note:

Superscript"−1"Refers to the power of functions, but not exponential power.

Properties of the inverse function:

1) The sufficient and necessary condition for the existence of the inverse function of the function is that the definition domain of the function is mapped one-to-one with the value range.

2) A function is monotonionic with its inverse function in the corresponding interval.

3) Most even functions do not have an inverse function (when the function y=f(x), 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 ).

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

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

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

6) The inverse function is reciprocal and unique.

Summary. The inverse of a function is a new function obtained by swapping x and y of the original function. Under what conditions does a function have no inverse function?

First of all, you have to understand what a function is. In layman's terms, a function takes every x, corresponding to only "one" y value.

The inverse of a function is a new function obtained by swapping x and y of the original function. Under what conditions does the one-high split-rent merge function have no inverse function? First of all, you have to understand what a function is.

In layman's terms, the function is to take each x, which only corresponds to the y value of "one Qi type jujube".

We can't say that a single function is an inverse.

It can only be said that the two functions are inverse to each other.

For example, the inverse function of y equals x is y equal to x

I can't say that y equals x is an inverse function.

In other words, the slow function where y is equal to x+1 is that y is equal to x-1

It can be said that the two of them are inverse functions of each other.

But it cannot be said that y equals x plus 1 is an inverse function.

As long as it is a one-to-one mapping, there is an inverse function.

The primary function y=kx+b has an inverse function, and the quadratic function y =ax 2+bx+c does not, because y=x 2, when y=1, x=1 or -1, y corresponds to 2 x's, not a one-to-one mapping The sufficient and necessary condition for the existence of an inverse function is that the definition domain of the function and the value range are one-to-one mapping; A function that strictly increases (decreases) must have an inverse function that strictly increases (decreases) [theorem for the existence of inverse functions].

There must be no inverse function for general even functions (but there is an inverse function for a special even function, e.g. f(x)=a(x=0), its inverse function is f(x)=0(x=a), which is a very special function), and there is not necessarily an inverse function for odd functions. There must be no inverse function about y-axis symmetry. If an odd function has an inverse function, its inverse function is also an odd function.

A function that strictly increases (decreases) must have an inverse function that strictly increases (decreases) [theorem for the existence of inverse functions].

dy=(df/dx)dx。

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-1(x). The condition for the existence of an inverse function is that the original function must be one-to-one (not necessarily within the entire number field).

1. Value range: The range of values that change due to the change of the dependent variable is called the value range of the function, which in the modern definition of function refers to the set of all the corresponding images of all the elements in the definition domain under a certain corresponding law.

2. In the function, the value range of the independent variable is called the definition domain of the function. For example, the definition range in y=ax+bx+c is the range of values of x.

3. The inverse function f(x) is symmetrical with respect to the straight line y=x; The graph of the function and its inverse function is symmetrical with respect to the straight line y=x, and the important condition for the existence of the inverse function is that the definition domain of the function is a mapping with the value domain; A function is monotonionic with its inverse function in the corresponding interval.