Symmetry of functions What is the symmetry between F (1-x) and f (x-1)? What is the inverse function of F (x + 2)? I'm confused now. Is it symmetrical with its inverse function y = X-2 or y = x?

Symmetry of functions What is the symmetry between F (1-x) and f (x-1)? What is the inverse function of F (x + 2)? I'm confused now. Is it symmetrical with its inverse function y = X-2 or y = x?

If f (1-x) = f (x-1), f (x) is symmetric with respect to y = 0
Let X - 1 = t, then f (T) = f (- t), even function
PS: if f (1-x) = f (x + 1), f (x) is symmetric with respect to y = 1. It can be considered that x is a distance. According to f (1-x) = f (x + 1), the function values corresponding to two points of distance X are equal
(PS: below ^ denotes - 1.)
Let f (x + 2) = y, then f ^ (y) = x + 2
X = f ^ (y) - 2, XY,
y=f^(x)-2
It is symmetric to its inverse function and y = x because it is an inverse function property
Note: the key is not x + 2, but the meaning of F (). F () represents a corresponding relationship. No matter what it is, the corresponding relationship remains unchanged. At most, it is just the change of definition field and image position
Therefore, asking the inverse function of F (x + 2) is actually asking the inverse function of F (T)
I haven't done this kind of problem for a long time. Don't be surprised if I make a mistake^_ ^)

How to judge the symmetry of function?
It is known that the increasing function y = f (x) defined on R satisfies ① f (x) = f (2-x); ② when x is greater than or equal to 0 and less than or equal to 1, f (x) = x square
Ask (1) to find the value of F (5.5)
(2) It is proved that when x belongs to R, f (x + 2) = f (x)
When the answer is on demand, it is said that because f (x) = f (x + 2), the image of y = f (x) is symmetrical with respect to the straight line x = 1! What's the matter! I hope that the person who answers me will not only tell me the answer, but also explain it here!
dial the wrong number! yes
Because f (x) = f (2-x), the image of y = f (x) is symmetrical about the line x = 1! What's going on! I hope the person who answers me not only tells me the answer, but also explains it here!

First of all, let me tell you why f (x) = f (2-x) is symmetric about a straight line. Let's ignore 2 instead of looking at f (x) = f (- x). This is an even function. We know that the even function is symmetric about the Y axis, that is, about x = O symmetry. We can be sure that f (x) = f (2-x) is symmetric about a straight line, that is, x = a symmetry, X about a symmetry

What is the image of an odd function symmetric about the origin

The origin symmetry means that the image is rotated 180 ° around the origin, and the new image completely coincides with the original one