The formula of function image symmetry and periodic function 1. On the symmetric function, what I want to ask is f|x + a|, which is about x =? What about symmetry? I remember symmetry seems to have a formula, I can't remember. 2. F (x) = loga | x + 1 | this function is symmetric with respect to x = - 1. I drew it with images. But how to express it in function? Is f (x + 1) = f (- x-1)? 3. I'd like to know about periodic functions. F (x + a) = f (x-a) is this a periodic function? What is the cycle? There seems to be a formula for periodic function. Find that formula. 4. Can f (x) = f (x + 1) deduce f (x + 1) = f (x + 2)? =============== Every question is ten. >

The formula of function image symmetry and periodic function 1. On the symmetric function, what I want to ask is f|x + a|, which is about x =? What about symmetry? I remember symmetry seems to have a formula, I can't remember. 2. F (x) = loga | x + 1 | this function is symmetric with respect to x = - 1. I drew it with images. But how to express it in function? Is f (x + 1) = f (- x-1)? 3. I'd like to know about periodic functions. F (x + a) = f (x-a) is this a periodic function? What is the cycle? There seems to be a formula for periodic function. Find that formula. 4. Can f (x) = f (x + 1) deduce f (x + 1) = f (x + 2)? =============== Every question is ten. >

Oh, my little brother, it seems that you are poor at learning functions!
1. A symmetric function has a formula: F (x) = f (A-X), which is symmetric about x = A / 2. As long as you see X and - x in an equation, it is a symmetric function. The axis of symmetry is that x equals the sum in brackets divided by 2. For example: F (1 + x) = f (3-x), then the axis of symmetry is x = (1 + X + 3-x) / 2 = 2, It can be written as f (x) = f (10-x) or F (5 + x) = f (5-x)
2. The function is about x = - 1 symmetry, which involves a specific function. You can first see that f (x) = loga | x | this function is an even function, f (x) = f (- x), about the y-axis symmetry, the symmetry axis is x = 0, f (x) = loga | x + 1 | that is, if the function f (x) = loga | x | is translated one unit to the left, then the symmetry axis is also translated one unit to the relative. It is concluded that about x = - 1 symmetry, the abstract function is f (x) = f (- 2-x) or F (X-2) = f (- x), As long as you are willing, you can write countless, according to the needs of the topic
3. The period of your function is | (x + a) - (x-a) | = 2A. The formula is f (x) = f (x + T), and the period is t. you should have a deeper understanding of periodicity combined with trigonometric functions
4. Yes, it is determined by the nature of the function itself. If x + 1 is replaced by X in the first equation, the second equation will be obtained. This is the reason why countless equations can be obtained. This is the basis for understanding the function, and many formulas use its derivation
Function this place is still very abstract, I suggest you ask your teacher more, Jinbang nomination!
Sorry, I just changed it. It's troublesome to call by mobile phone. It's easy to make a wrong number
For you this question, today specially comes to the Internet, if feels satisfied must praise
It can't be said that they are equivalent. Firstly, the expression of their specific functions is different. Secondly, the domain of definition of functions is also different
F (x) = loga | x + 1 |, its domain of definition is x not equal to - 1, so to ensure that the range in brackets of abstract function is not equal to - 1, for example, in F (X-2) = f (- x), the domain of definition is x not equal to 1. From the first equation to the second equation, we only need to replace X in the form with X-2
Next, I'll explain the replacement with specific functions
F (x) = x + 1 (domain x > 2) replace its X with x-3
F (x-3) = (x-3) + 1 = X-2
You only need to understand x as the symbol of F (), and it's better to understand it with the concept of composite function and translation thought
You see, their expressions are different, and their domains are also different
Function is the most abstract part of high school mathematics, which must pass
Take a good look at my answers