If there are two points A(x 1, y 1) B(x2, y2), the coordinates of their midpoint p are ((x 1+x2)/2, (y 1+y2)/2). The symmetry point of any point of (x, y) about (a, b) is (2a-x, 2b-y).

Then (2a-x, 2b-y) is also on this function. There is f(2a-x)= 2b-y transposition, and y=2b- f(2a-x).

The image of the function is symmetrical about the point (a, b). From the above expansion, we can know that the symmetry point of any point (x, y) on this function about (a, b) is (2a-x, 2b-y).

Then (2a-x, 2b-y) is also on this function.

There is f(2a-x)= 2b-y transposition, and y=2b- f(2a-x).

Note that here y can be regarded as f(x).

So, to sum up, if the image of a function is symmetrical about point (a, b), then the relationship that this function should satisfy is f(x)=2b- f(2a-x).

The midpoint formula is a special case of the fixed point formula. Using the midpoint formula, we can find the coordinates of two points on the plane and solve a kind of symmetry problem about a point.