Confusion about the understanding of inverse functions in higher mathematics

Not contradictory. The image of the inverse function is correct with respect to the y=x-axis symmetry of the straight line. It's both. For example, functions.

y=f(x)=x+6.Swapping x and y gives x=f(y)=y+6 is its inverse function, or deformation:

x= (y)=y-6, in this case, x is regarded as the dependent variable and y is regarded as the independent variable, that is, x is the original y in the mind, which is the inverse function. Otherwise, x=y-6 is the same function as the original function.

x= (y) Draw the image as y is still the ordinate axis, x is still the abscissa axis, the key is to understand it from the perspective of mapping.

In y=f(x) and x=(y), x and y remain unchanged and are still taken from the set of coordinate points. f represents an x-to-y map, just the reverse of the map. Represents a reflection of f.

So x is still a point on the abscissa axis, and y is still a point on the ordinate.

Y= (x) retains the mapping, and swaps the set of x,y relative to x= (y), so its image is symmetrical with respect to y=x.

The function y=f(x), then let its inverse function be x= (y), and exchange x and y to get y= (x)Isn't it just swapping the x,y axis, of course, about y=x symmetry. That's what the inverse function is all about.

To put it simply, a function is a mapping relationship (f) between the set of independent variables and the set of dependent variables, and the inverse function is a new mapping relationship ( ) obtained by swapping the definition domain and value range of the direct function

Thus, the inverse of the function y=f(x) can be expressed as x= (y) or y= (x). The difference between x and y in the inverse function of the two expressions is only the difference in the notation of the independent variable and the dependent variable. The images of x=(y) and y=f(x) are the same image, because here you represent one of the two different functions as the dependent variable of the other in the same coordinate system, which naturally coincides with the direct and inverse functions [conversely, the two identical functions x=2y (independent variable y) and y=2x (independent variable x) have different images in the same coordinate system].

So in order to distinguish this pair of functions in the same coordinate system, we usually express the inverse function as y= (x).

（1-y）/（1+y）

1+x）y=1-x，xy+x=1-y,x（1+y）=1-y,x=（1-y）/（1+y）

x/4y=4x→x=y/4→y^(-1)=x/4

e^（sin²x）

y=e^（sin²t）=e^（sin²x）

y=2^x/（1+2^x）

2^y(1-x)=x，2^y-2^y*x=x，（1+2^y）*x=2^y，x=2^y/（1+2^y）

When writing the inverse function here, you can't swap x and y. That is, the inverse function of y=arctanx is x=tany.

So x=tany, dx dy=(tany).'=sec2y

Dy dx=1 (dx dy)=1 sec2y=cos2y=cos2y (cos2y+sin2y)=1 (1+sin2y cos2y)=1 (1+tan2y)=1 (1+x2)

The words are ugly and small, please be patient.

The process to is as follows:

For. (e^x)² 2ye^x - 1 = 0 ……Yes recipe, gotcha.

e^x)² 2ye^x + y² =1 + y²(e^x - y)² 1 + y²

Further down, easy to get, you should know!

If you think of exp(x) as x, Eq. 1 becomes a quadratic function, and it is obvious to use the quadratic function to find the root formula.

The inverse function of higher mathematics is found like this:

1. The method of finding the inverse function: let the definition domain of the function y=f(x) be d, and the value range is 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).

From this definition, it can be quickly concluded that the definition domain d and the value range f(d) of the function f are exactly the value range and definition domain of the inverse function f-1, and the inverse function of f-1 is f, that is, the functions f and f-1 are inverse functions of each other. ArcCOS is calculated using the following formula: COS (arcsinx) = 1-x 2).

2. The symbol of the inverse function is denoted as f-1 (x), and in Chinese textbooks, the inverse trigonometric function is denoted as arcsin, arccos, etc., but in some countries in Europe and the United States, the inverse function of sinx is denoted as sin-1 (x).

An inverse function is a function that does the inverse of a definite function.

Generally speaking, if the domain of the function y=f(x)(x a) is c, if we find a function g(y) where g(y) is equal to x, then the function x= g(y)(y c) is called the inverse function of the function y=f(x)(x a), denoted as y=f (-1)(x), and the inverse function x=f (-1)(y) defines the domain and the range of the function y=f(x) respectively.

The most representative inverse functions are logarithmic and exponential functions.

Because it's an inverse function. The inverse function value at 0 is actually the value of the argument variable when the original function value is 0, which is 1 (f(1)=0).

However, the analysis of this topic does not seem to be rigorous, because the analysis is actually aimed at (f'1, that is, the inverse function of the derivative of the original function, but the question requires the derivative of the inverse function, and the two are still a little different.

Answer: The formula for finding the derivative of the inverse function:

Therefore, the one who wants the original question.

f^（-1）‘ 0） =1/ f'(1) =2 2 where: f(1) =1,1) 1+t ) dt =0

The function is actually a correspondence between two sets of numbers, and the inverse function is actually based on the original function, which does not change the correspondence between the two sets of numbers, but only changes the position of the corresponding two sets: the original is x1 y1, x2 y2, ......Now it's Y1 x1, Y2 x2 ......

The former is the original function, and the latter is the inverse function – this is one way to express the function: the enumeration method. It can be seen that the "definition domain" and "value range" of the inverse function are swapped with the original function.

As you can imagine, not all functions have original functions. Functions allow for many-to-one relationships, but not one-to-many. Therefore, all functions with inverse functions are "one-to-one" relationships.

It can be simply understood that the number of elements in the "definition domain" and "value field" of the function is equal and can be paired one by one.

Suppose the function y = f(x) (the standard notation for this function is: f:x y) has the inverse function:

y→x。Then, the function image f of f and the function image w of f must satisfy the following relation:

The point (x,y) is on f, if and only if the point (y,x) is necessarily on w.

Obviously, these two points are symmetrical about the straight line y = x. When axisymmetric points can be found on w for all points on f, f and w are themselves axisymmetric, and that's exactly what happened.

Finally, two images of axisymmetry, necessarily "identical".

Their images are symmetrical about y x. Does it mean that the shape is the same but the direction and position are different? For example, the image of y x 2 (x 0) is a half parabola with the opening upward, and its inverse function is that the opening is like the half parabola on the right, and it is the same size.

If there is an inverse function for the function, then the inverse function can be drawn according to the symmetry about y=x, and the domain of the inverse function is the domain of the original function, and the domain of the value is the domain of the original function. It's easy to understand after drawing the image. But not all functions have inverse functions!!

The square of y=x upstairs has no inverse function (I don't know how he learned math). And what you said is that the inverse function has the same graphical tendency as the original function (monotonicity).

(1) 1-2x=e y x=(1-e y) 2 So the inverse function is y=(1-e x) 2,x>0 (2) y=2cos(x 2) y 2=cos(x 2) x 2=arccos(y 2) x=2arccos(y 2) so the inverse function is y=2arccos(x 2) -2<=x<=2

In general, if the corresponding f of the determined function y=f(x) is a one-to-one correspondence from the definition domain of the function to the value domain, then the function determined by the "inverse" of f corresponding to f-1 is called the inverse function of the function, and the definition domain and value range of the inverse function x=f-1(x) are respectively the value domain and the definition domain of the function y=f(x). Here's an example: Original Form:

y=(2x-3) (5x+1) x belongs to r and x≠-1 5 solution: y(5x+1)=2x-3 5xy+y=2x-3 x(5y-2)=-y-3 x=-(y+3) (5y-2) Swap x and y to get the inverse of the original function: y=-(x+3) (5x-2) (x≠2 5) General solution of the inverse function:

1. Solve x from the original function equation, that is, use y to represent x 2. Replace all x with y, and replace all y with x to get the inverse function The properties of the reaction function in middle school are mainly as follows: (generally explicit function) (1) The image of two functions that are inverse functions of each other is symmetrical with respect to the straight line y x; (2) The sufficient and necessary condition for the existence of an inverse function is that the function is monotonic in its defined domain; (3) a function is monotonionic with its inverse function in the corresponding interval; (4) There must be no inverse function for even functions, and there is not necessarily an inverse function for odd functions. If an odd function has an inverse function, its inverse function is also an odd function.

There are explicit and implicit functions, and the inverse function of the explicit function has all the above properties. Implicit function properties: (1) All implicit functions have inverse functions; (2) the image of two functions that are inverse functions of each other is symmetrical with respect to the straight line y x; (3) The monotonicity of a continuous function is consistent within the corresponding interval.

What's the point of nothing?