What are the knowledge points of compound functions in high school? Let s be a freshman in high scho

The f(x) range should be discussed on a case-by-case basis, whether it is greater than 0 or less than 0. Then apply f(x) to the corresponding function. See**.

Known step function: f(x)=1+x, (x<0)...f(x)=x，(x≧0)..Find f[f(x)]=

Solution: f[f(x)]=1+f(x)=1+1+x=2+x, when f(x)=1+x<0, i.e. x<-1];

.From: f[f(x)]=f(x)=x, [when f(x)=x 0, i.e. x 0];

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f(2x+1)=x²-2x

Requirement f(2).

Then let 2x+1=2

x=1 2So(2)=(1 2) -2 1 2=-3 4f(x)+2f(1 x)=3x

Then let x=2f(2)+2f(1 2)=6 (1).

Let x=1 2

f(1/2)+2f(2)=3/2 (2)

3f(2)=-3

f(2)=-1

2f(x)+f(-x)=3x+2

Let x=2 then 2f(2)+f(-2)=3 2+2

f(-2)=-16/3

So f(2)=20 3

Requirement f(2).

So the key to these questions is to make 2 in parentheses after f, and if you find f(x).

Then x and 1 x and x and -x can be substituted in the last two.

For example, let x=1 x, and get a new functional equation, which can be solved with the original coupling.

The first one directly substitutes x for half to get f(2), and the second one replaces x for 1 x, remember, and the x in the next 3x also needs to be changed! Then subtract the two formulas to get the analytic formula of the function! Substitute to get the answer!

For the third one, just replace x with 2!

To do this kind of question, the main thing is to know the overall substitution! After changing, there is no way to form the original form! But some can be opportunistic! Such as the first one! Haha, it's actually better to practice more! Think more! Finally, good luck!

f(x) is x instead of x in f(x), so f(x)=2x -1 In the same way, f]g(x)] is to replace x in f(x) with 1 x 1 f[g(x)]=2 (x 1)-1 g[f(x) 2].

(1) Composite function y=f(u),u=g(x)When it is not compounded, the domain of the function y=f(u) is the range of values of u, and after compounding, the domain of the function y=f[g(x)] is the range of values of x. However, the value range of the inner function should not exceed the defined domain of the outer function.

My approach to this question is, 0 x 1,===>-1 x-1 0.This finds the domain of the function y=f(x) as [-1,0].∴1≤x+2≤0.

=>-3≤x≤-2.The domain of the function f(x+2) is [-3,-2].(2) f(x+2)=1 f(x)

=>f(x+4)=1/f(x+2)=f(x).The function f(x) is a periodic function with period 4. f(5)=f(1)=-5.

As can be seen from the foregoing, f(-3)=1 f(-5)f(1)=f(-3).∴f(-5)=1/f(-3)=1/f(1)=-1/5.

f[f(5)]=f(-5)=-1/5.

1.x-1 and x+2 have the same value range, and the definition field refers to the range of x, so x is different. 2.

Medium can know that f(x) is a periodic function. In the same equation, the same unknown must be the same. I don't know if you understand?

The domain of f(x-1) refers to the range of x, and the two x's are not the same.

f(x) is x instead of x in f(x).

So f(x)=2x -1

In the same way, f]g(x)] is to replace xf[g(x)]=2 (x +1)-1 in f(x) with 1 x +1

g[f(x)+2]

f(2x+1)

1/[(2x+1)²+1]

1/(4x²+4x+2)

As can be seen from the title:

1. f(t)={t-5)^2+625, 0<=t<10(t-35)^2-25, 10<=t<=202.f(t)max=f(5)=625 when 0<=t<10, and f(t)max=f(10)=600 when 10<=t<=20

Because 625>600, the maximum value of f(t) is 625

However, I want to make it clear that this type of problem is a piecewise function, not a composite function.

I understand, it's not that your answer is wrong, but that the range of 10<=t<=20 is that you substituted 20 into the wrong one, and you should substitute ten into it!!

The monotonic interval is the behavior on the sub-interval of the defined domain, and you find the defined domain to be [-4,2], which is good.

The outer function is the power function y= u=u (1 2), in [0,+ monotonically decreasing, the quadratic function of the inner function u=-x 2-2x+8=-(x+1) 2+9, the opening is downward, and the axis of symmetry x=-1 satisfies the defined domain, so in (-4,-1) monotonically decreasing, (-1,2) monotonically decreasing, known by the same increase and difference of the composite function, (-4,-1) monotonically decreasing, (-1,2) monotonically decreasing.

Separated according to the monotonic zone of the intrinsic function.

Self-math teacher's suggestion: You can't blindly brush up on math questions, in fact, the method is very simple! Help your child improve their score quickly and sign up first.

As a simple example, is f(x)=x +2x+1 the same as f(x) monotonic performance under the root number?

For example, y = u 2 , u = 2x+1, originally y is a function of u, and u is a function of x, substituting y = (2x+1) 2 , it becomes a function where y is x, and this function is called a composite function, which is a composite of the above two functions.

In addition to addition, subtraction, multiplication, division, multiplication, square, logarithm, etc., there are also compounds.

At this stage, the people's education version of high school mathematics does not specifically mention this concept, and there are no corresponding practice questions in the book. The teacher will add to the monotonicity of the logarithmic function.

A function of the form y=f(g(x)) is a composite function, which is a composite of two functions, u=g(x) and y=f(u). It's like two dough kneaded into one dough.

For example, the function y=l0g2(3x+4), which is a compound of y=log2u, u=3x+4, the first function u is the independent variable, y is the function, the second function x is the independent variable, and u is the function.