Questions about math functions in high school, math high school function questions

f(x) satisfies f(x+3)=-f(x), i.e., f(x) is a periodic function, and the period t=6 (if f(x+3)=f(x) period is x+3-x=3, and f(x+3)=-f(x) period t=2 3=6) f(2012)=f(2) (2012=2010+2=335 6+2) f(2)=f(-1+3)=-f(-1)=-1=f(2012) 2. f(x+2)=-1 f(x) f(x) is a periodic function with t=8 as the period (f(x+2)=f(x)period=x+2-x=period 2=2 2=, and then period 2=4 2=8) f( and f(x+2)=-1 f(x) i.e. f( Note: please be good that f(x) is an even function, when x is greater than or equal to 2 and less than or equal to 3 f(x)=x condition.

f（x+3)=-f(x)

Substituting x-3 instead of x gives f(x)=-f(x-3), and then f(x+3)=-f(x) gives f(x)=-f(x+3)-f(x-3)=-f(x+3).

f(x)=f(x+6) period is 6

f(2012)=f(335*6+2)=f(2)f(x)=-f(x+3)

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

f(2012)=-1

Questions about f(x+a)=-f(x) can be done in the same way.

Summary. First, the function f(x) has a c relationship with g(x).

Please send me the question** to see!

This question, thank you, the first and second questions are written, first give me the third question, the teacher can write it or not, thank you.

This finale question is troublesome and time-consuming to answer.

That's fine. Well, the first and second questions, write a question and send it to me, try to hurry, thank you.

I'm calculating.

First, the function f(x) has a c relationship with g(x).

The second question is that the function f(x) does not have a c relationship with g(x), then the value range of a is (-2 2,0)u(0,2 2).

This question a=0 is true.

Second, if the function f(x) and g(x) do not have a c relationship, then the value range of a is (-2 2, 2 2).

Simple. y=g(x+m)=2sin(4x+4m-2 3) obviously knows that y=sinx is an odd function, so.

When 4m-2 3=0, the function y=g(x+m) is an odd function.

i.e. m = 6

Or when 4m-2 3=2k (k is an integer), the function y=g(x+m) is also an odd function. Trembling.

Hope. Xie Zhu Xie argued.

There is nothing I understand.

**Friends. Answer them one by one.

2sin4x is a strange letter or waiter, substituting x=x+m so that 4m-2 3=0

One of its solutions is V6, considering that it is a periodic function, plus K2V.

The answer is 6+k 2 vultures

Solution: f can be derived from the problem'(x)=8 (2-x)-6 (x-1), let f'(x)=0, we get x=34 25

Order f'(x) x=(1,34 25) of 0 is monotonically decreasing at that point and then letting f'(x) 0 is found for x=(34 25,2), i.e., monotonically increasing at this point.

So when x=34 25 the function has a minimum value.

The value is f(x)=5

The maximum value is one or both of the two endpoints, because f(1)=7=f(2)=7, so when x=1 or 2, f(x) has a maximum value, and the value is 7

Because f(x+6)=-1 f(x+3)=f(x), so f(x) period is 6, and 2010=335*6+0 so f(2010)=f(0).

0 [0,3] has f(0)=1

The questions are incomplete, how to answer!

is a checkmark!! You search the Internet, and it's derived from the mean theorem, and I don't know if you've learned it or not.

I've got a diagram for you, where you bend and curve smoothly! The y-axis and y=x are asymptotes of f(x)=x+1 x!

Therefore, when it is greater than 0, at (0,1] is the single-point subtraction interval (1,+ is the monotonic increase interval, x=1 obtains the minimum value, and the minimum value y=2

Have you ever studied derivatives? If you have learned, it will be easy to do.

f'(x)=1-1/xx

It can be seen that at (0,1) f'(x)<0, so (0,1) is monotonically decreasing when the interval is (1,+00) f'(x) >0, so (1, +00) is a monotonic increase interval. The idea is clear.

There is no need for a fishhook function, which is used after knowing its monotonicity, not by judging its monotonicity.

I drew you a diagram of the hook function, smooth curves when you bend the place! The y-axis and y=x are asymptotes of f(x)=x+1 x!

Therefore, when it is greater than 0, at (0,1] is the single-point subtraction interval (1,+ is the monotonic increase interval, x=1 obtains the minimum value, and the minimum value y=2

Generally, it is with absolute value, and it is easy to combine numbers and shapes, but it is best to use the discussion method when answering questions, because the combination of numbers and shapes cannot be well displayed to the marking teacher. For example, y=|x-2|-|x-8|

The mean inequality is generally defined as a+b-squared 2ab and the variant a+b 2ab

When using, pay attention to the equality of one, two, and three.

That is, for a+b 2ab, a, b are positive.

When their sum is a fixed value, the product has the greatest value. When the product is a fixed value, and there is a maximum value.

Three-equality means that you need to test after applying the mean theorem, because when a = b, you can take the equality sign of the inequality, and you have to test whether a and b can be equal.

This is the most basic, the mean theorem is an important knowledge point in high school mathematics, and there are many variants and extensions, which is a compulsory knowledge point for the college entrance examination.

According to the title. 1) When the pollution fraction of lake water is constant, it means that the function g(t) does not change with the change of t, that is, g. g'(t)=0

Find the derivative of the function g(t), i.e., g'(t)=(r/v)[g(0)-p/r]e^(r/v)t=0

So g(0)=p r

2), when g(0) p r, g'(t) <0, the function g(t) decreases monotonically.

The results indicated that with the increase of time, the pollution quality fraction of lake water decreased, and the pollution degree of lake water gradually improved.

It seems a bit of a problem to look at the stove: what about when x 0? , The topic should not be scattered and complete, I can only tell you that this type of discussion should be discussed by classification:

When x 0, the former is 0 and the latter is 0

When x 0, the former is 0 and the latter is 0

When x 0......

In summary, f(x) 0