Function y = f (x), function g (x) is the inverse function of F (x), the image of H (x) and G (x) is symmetric about the line y + x = 0, then H (x)=

Let (x, y) be any point on the H (x) image: then y = H (x)
And (- y, - x) is the point on G (x) image, then: - x = g (- y);
The function g (x) is the inverse function of F (x), so (- x, - y) is on the image of y = f (x)
So: - y = f (- x), that is: y = - f (- x)
So: H (x) = - f (- x)

If the images of the functions y = f (x) and y = g (x) are symmetric with respect to the line y = x, then the inverse function of the function y = f (2x) is

0

There is no counterexample. As long as two functions are inverse functions, they must be symmetric about y = X. because in the process of finding inverse functions, we have made an exchange between X and y, and this exchange is actually to find the function that the original function is symmetric about y = X

What pairs of symmetries about the image of the function y = f (x + 2) and the inverse function f (x + 2)

Method 1: let t = x + 2, then f (x + 2) = f (T), f * (x + 2) = f * (T), (Note: f * denotes the inverse function) is easy to know, the images of y = f (T) and y = f * (T) are symmetric with respect to the straight line y = t, and the images of y = f (x + 2) and y = f * (x + 2) are symmetric with respect to the straight line y = x + 2

Are two functions symmetric about y = x, must they be inverse functions?

If it is symmetric about y = x everywhere, then it is the inverse function
Otherwise, as long as there is a little asymmetry, it is not (⊙ o ⊙) oh

The image of the function f (x) defined on R is centrosymmetric about the point (1,2), and there is an inverse function f - (x) in F (x). If f (4) = 0, then f - (4) is many In addition, the point P is the moving point on the line 2x + y + 10 = and the straight lines PA and Pb tangent to the circle x + y = 4 at two points a and B, then what is the minimum area of the quadrilateral paob (o is the origin)?

According to f (4) = 0, the point P (4,0) must be on the function image. According to the problem, the symmetry point Q (- 2,4) of point P (4,0) about point (1,2) is also on the image of function 2, so f (- 2) = 4. According to the relationship between the corresponding value of function and inverse function, F - (4) = - 2

Let the image of function f (x) be symmetric with respect to point (1,2), and there is an inverse function F-1 (x), f (4) = 0, then F-1 (4) =? F-1 (x), denotes the inverse function of F (x)

From the symmetry of the image of the function f (x) with respect to points (1,2), we can obtain
F (x + 1) + F (1-x) = 4, which holds for any X
In the above formula, take x = 3 to get
F (4) + (- 2) = 4, i.e
f(-2)=4
Thus, F-1 (4) = - 2

Let the graph of function f (x) be symmetric with respect to point (1,2), and there exists inverse function F-1 (x), f (4) = 0, then F-1 (4)=______ .

From the symmetry of the image of the function f (x) with respect to points (1,2), we can obtain that f (x + 1) + F (1-x) = 4. For any x, it holds in the above formula,
Taking x = 3, we get f (4) + F (- 2) = 4, and f (4) = 0
∴f（-2）=4∴f-1（4）=-2
Therefore, fill in - 2

Let the graph of function f (x) be symmetric with respect to point (1,2), and there exists inverse function F-1 (x), f (4) = 0, then F-1 (4)=______ .

Let the image of function f (x) be symmetric with respect to point (1.2), and there is an inverse function f (x) - 1, f (4) = 0, then f - 1 (4)=___ ?

The image of function f (x) is symmetric about point (1.2); if the image of function f (x) is symmetric about point (a, b), then f (2a-x) + F (x) = 2B; f (4) + F (- 2) = 4; therefore, f (- 2) = 4; that is, if f (x) passes through (- 2,4); then its inverse function passes (4, - 2); that is, F-1 (4) = - 2;
For adoption

Function y = f (x), function g (x) is the inverse function of F (x), the image of H (x) and G (x) is symmetric about the line y + x = 0, then H (x)=