Symmetry of function is a basic property of function. Symmetry not only exists widely in mathematical problems, but also can be used to solve problems more simply. Symmetry also fully embodies the beauty of mathematics.

1 and the symmetry of function y = f (x) (itself);

(1) Theorem 1: The image of the function y = f (x) is symmetric about the straight line x=(a+b)/2;

→ f (a+x)= f (b-x)→f (a+b-x)= f (x)

Special ones are:

① the image with function y = f (x) is symmetrical about the straight line x=a → f (a+x) = f (a-x) → f (2a-x) = f (x);

② the image of function y = f (x) is symmetric about y axis (odd function) → f (-x) = f (x);

③ The function y = f (x+a) is an even function →f (x) is symmetric about x=a;

(2) Theorem 2: The image of function y = f (x) is symmetric about point (a, b);

→f(x)= 2 B- f(2a-x)→f(a+x)+f(a-x)= 2b

Special ones are:

① the image with function y = f (x) is symmetrical about point (a, 0) → f (x) =-f (2a-x);

② the image with function y = f (x) is symmetrical about the origin (odd function) → f (-x) = f (x);

③ the function y = f (x+a) is odd function →f (x) is symmetric about point (a, 0).

Theorem 3: (Properties)

(1) if the image of function y=f (x) has two vertical symmetry axes x=a and x=b(a is not equal to b), then f (x) is a periodic function, and 2|a-b| is one of its periods;

② If the image of function y=f (x) has a symmetry center M(m, n) and a vertical symmetry axis x=a, then f (x) is a periodic function, and 4|a-m| is its period;

③ If the image of function y = f (x) is centrosymmetric (a≠b) about points A(a, c) and B (b, c), then y = f (x) is a periodic function, and 2 | a-b | is a period of it;

(4) If the inverse function of a function is itself, then its image is symmetric about the straight line Y = X. ..

2. Symmetry of two function images:

(1) The images of function y = f (x) and function y = f (-x) are symmetrical about the straight line x=0 (that is, the Y axis);

(2) The images of function y = f (mx-a) and function y = f (b-mx) are symmetrical about the straight line x=(a+b)/2m;

In particular, the images of y = f (x-a) and function y = f (a-x) are symmetrical about the straight line x=a;

(3) The analytical formula of the symmetry of the image of the function y = f (x) about the straight line x=a is y = f (2a-x);

(4) The analytical formula of the symmetry of the y = f (x) image of the function about point (a, 0) is Y =-f (2a-x);

(5) the images of functions y = f (x) and a-x = f (a-y) are symmetrical about the straight line x+y = a. ..

Images with functions y = f (x) and x-a = f (y+a) are symmetrical about the straight line x-y = a.

The image of function y = f (x) and the image of x = f(y) are symmetrical about the straight line x = y.