The problem of function image, the problem of function image symmetry, please write the proof in detail, 1. What is the symmetry between the image of function y = f (x-1) and the image of function y = f (1-x) 2. What is the symmetry between the image of function y = - f (x-1) and the image of function y = f (1-x) The first one I think is about x = 1 symmetry. I got it by enumerating. I made f (x) = x, X & # 178;, 1 / X try to get it. I know how to prove it first Second, I gave some examples, but I didn't get them. I also want to prove them
1、
Let t = X-1,
Then 1-x = - T
Because the image of y = f (T) and the image of y = f (- t) are symmetric with respect to t = 0
So the image of the function y = f (x-1) and the image of the function y = f (1-x) are symmetric with respect to 1-x = 0, that is, x = 1
2、
As above, let t = X-1
Because the image of y = f (T) and the image of y = - f (- t) are symmetrical about the origin (t = 0, y = 0)
So the image of the function y = - f (x-1) and the image of the function y = f (1-x) are symmetric with respect to point (1,0)
What is the symmetry of the image of the function y = sin (x + 3 Π / 2)?
y=sin(x+3π/2)=-cosx
[odd variable, even constant, sign in quadrant]
So the axis of symmetry is x = k π [the point where y = + - 1]
Given the function f (x) = the x power of a plus (x + 1) parts (X-2) (a > 1), we prove that f (x) is a function on (- 1, positive infinity)
f(x)=a^x+(x-2)/(x+1)
f'(x)=a^x(lna)+3/(x+1)^2
When x > - 1, f '(x) > 0, so it is (- 1, positive infinity) monotone increasing