A high school function problem do somebody a favour, Y ^ 2 = - x ^ 2 + 4ax-3 x ∈ [1,2] to find the range,

A high school function problem do somebody a favour, Y ^ 2 = - x ^ 2 + 4ax-3 x ∈ [1,2] to find the range,

First of all, f ^ 2 (x) BF (x) C = 0 may have two or one different real roots. First of all, we need to make an image of the function f (x), and then make a straight line parallel to the X axis, y = m, intersecting with the image

Ask an expert to help you solve a math problem (function),
It is known that F: R + → R, f (x) = logx; G: s → R, G (x) = 2-x
Try to define s to make fog meaningful

Fog is the meaning of compound function
fog=log(g(x))=log(2-x)
2-x is defined on R +, so 2-x > 0, X

The method of substitution of higher one function
For example: F (x + 1) = x & sup2; - X - 5, find the analytic expression of F (x)
Let x + 1 = t, then x = t + 1, f (T) = (T + 1) & sup2; - T-6, and then change t into x ······ I don't understand. If it's not right, please correct it for me and write the positive solution. If it's right, please tell me why t can be changed into x directly

In fact, x = T-1 is derived from x + 1 = T. The so-called substitution method is to replace x with another element symbol, and then change the other element back to X after a series of modifications

[if f (2x + 1) = x ^ 2-4x + 2, find f (3-4x)]
If f (2x-1) = 4x ^ 2 + 4x + 2, then the analytic expression of F (x)
If f (2x) = 4x ^ 2 + 1, the analytic expression of F (x) is
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Note: it's not the answer to these questions, it's the same type of questions as those above!
The more, the better!
If it's good, I'll add points!
Also, do you have any ideas to pay attention to or specially consider when doing these topics?
Seek expert guidance method experience! QAQ

You don't have to ask for the same topic
The solution of exchange problem is progressive
Can understand the following, master the exchange method
1. If f (x) = 4x ^ 2 + 1
Let u = x, then: x = u,
If all x in the original formula is replaced by u, f (U) = 4U ^ 2 + 1 -- ① can be obtained
Because of the habit of using X to express independent variables, the expression of ① is: F (x) = 4x ^ 2 + 1
2. If f (2x) = 4x ^ 2 + 1,
Let u = 2x, then: x = u / 2,
If all x in the original formula is replaced by u, f (U) = 4 (U / 2) ^ 2 + 1 = u ^ 2 + 1 -- ② can be obtained
Because of the habit of using X to express independent variables, the expression of ② is: F (x) = x ^ 2 + 1
3. If f (2x-1) = 4x ^ 2 + 1
Let u = 2x-1, then: x = (U + 1) / 2,
If all x in the original formula is replaced by u, f (U) = 4 [(U + 1) / 2] ^ 2 + 1 = (U + 1) ^ 2 + 1 = u ^ 2 + 2U + 2 -- ③ can be obtained
Because of the habit of using X to express independent variables, the expression 3 is: F (x) = x ^ 2 + 2x + 2
4. If f (2x-1) = 4x ^ 2 + 1, find f (3-4x)
F (U) = u ^ 2 + 2U + 2
Let: u = 3-4x, substitute:
f（3-4x）=（3-4x)^2+2（3-4x）+2
It can be simplified
It's not difficult to understand the exchange question, just step by step

There are four methods to find f (x),
The teacher said in class, but not very skilled, want to listen in detail, the best is the teacher answer, the best example

The function f (x) has three elements: definition field, value field and corresponding relation F. it is known that the corresponding relation f (x) of the function that can be expressed by analytic method satisfies some conditions, how to find the analytic formula F (x). There are many types of problems and many solutions. But the most common ones are three solutions

Why can we change f (T) directly to f (x) in the last step of the substitution method

If two functions have the same domain of definition and the same corresponding rule, then they are the same function, for example: F (T) = 3T + 2F (x) = 3x + 2. Their corresponding rules are: the third times of the independent variable plus 2; the domain of definition is the same, so they are the same function. Therefore, f (T) can be directly replaced by F (x)

Solving the analytic formula by the system of equations of higher one function
Example: given 2F (1 / x) + F (x) = x (x ≠ 0), find f (x)
I just read high one, these questions are very abstract, please explain in detail, prompt how to learn high one abstract function

2F (1 / x) + F (x) = x (x ≠ 0), ①, (x in this formula can be replaced by any number or letter except 0)
Let x = 1 / X be substituted into Formula 1, then 2F (x) + F (1 / x) = 1 / x, 2
② If you multiply formula 2 and subtract Formula 1, you can eliminate f (1 / x) and get 3f (x) = 2 / x-x = (2-x ^ 2) / X
So: F (x) = (2-x ^ 2) / 3x
Understand, this kind of questions are done in this way;
Hope to help you, if you do not understand, please hi me, I wish learning progress!

For example, given f (x + 2) = 4x + 7, find f (x) and so on

The key is to express 4x + 7 with x + 2. The breakthrough of this kind of problems is here. You will feel very simple when you understand this kind of problems
In other words, 4x + 7 = 4 (x + 2) - 1, first make up 4x, and then add a few more if you see how much less, so that the equal sign is equal to the left and right
Then f (x + 2) = 4 (x + 2) - 1
If x + 2 is regarded as a whole and replaced by T, then f (T) = 4t-1
Then change t to x, that is, f (x) = 4x-1

Solving function analytic formula by solving equations in senior one (please explain in detail)
af(x)+bf(-x)=cx;
AF (- x) + BF (x) = - CX; find f (x)

af(x)+bf(-x)=cx; 1
af(-x)+bf(x)=-cx; 2
Formula 1 * a
Formula 2 * B
We obtain a ^ 2F (x) + ABF (- x) = ACX 3
abf(-x)+b^2f(x)=-bcx 4
3-4 (a^2-b^2)f(x)=(ac+bc)x
So f (x) = CX / (a-b)

The solution of analytic formula of higher one function,
How to do f (x + 1) + F (x-1) = 2x square - 2x + 4?
Given that f (x) is a polynomial, f (x + 1) + F (x-1) = 2x square - 2x + 4, how to find the analytic expression of F (x)?

Let f (x) = ax ^ 2 + BX + C
f(x+1)+f(x-1)=a(x+1)^2+b(x+1)+c+a(x-1)^2+b(x-1)+c
=2ax^2+2bx+2a+2c=2x^2-2x+4
The equations 2A = 2 are obtained
2b=-2
2a+2c=4
The solution is a = 1, B = - 1, C = 1
So f (x) = x ^ 2-x + 1