Find the period of the function 15, how to find the period of the function?

f(x+a)+f(x)=f(x+a)f(x)f(x+2a)+f(x+a)=f(x+2a)f(x+a)f(x+a)f(x)+f(x+a)=f(x+2a)f(x+a)f(x+a)=0...Any constant is period, f(x)+1=f(x+2a).f(x+a)=1..

f(x)=f(x+2a) contradiction.

sin3x*cos3x=(2sin3x*cos3x) 2=(sin6x) 2, so period t=2 6= 3. Since the maximum and minimum values of sinux of the sinusoidal function are 1 and -1, respectively, when sin6x takes 1, the function obtains a maximum value of 1 2. When sin6x takes -1, the function obtains a minimum value of -1 2.

2）1/2-sin2x

The constant term does not affect the minimum positive period of the function, so the period t=2 2=. Since the maximum and minimum values of sinx of the sinusoidal function are 1 and -1, respectively, when sin2x takes -1, the function obtains the maximum value of 1 2-(-1)=3 2;When sin2x takes 1, the function obtains the minimum value of 1 2-1=-1 2. （3）y=sin(x-π/3)cosx

For this problem, we need to use the formula of product sum difference: sin cos = [sin( +sin( - 2

The expression is obtained by using the formula of product sum difference: y=sin(x- 3)cosx=[sin(2x- 3)+sin(- 3)] 2=[sin(2x- 3)-sin 3] 2=sin(2x- 3) 2- 3 2

So the period t=2 2=. Since the maximum and minimum values of the sinusoidal function sinx are 1 and -1, respectively, when sin(2x-3) takes 1, the function obtains the maximum value of 1 2-3 2, i.e., (1-3) 2. When sin(2x-3) takes -1, the function obtains the minimum value -1 2-3 2, i.e., (-1-3) 2.

strong answer supplement the third question simplification is a bit wrong, the correct one should be:

So the period t=2 2=. Since the maximum and minimum values of the sinusoidal function sinx are 1 and -1, respectively, when sin(2x-3) takes 1, the function obtains the maximum value of 1 2-3 4, i.e., (2-3) 4.

Can it solve your problem?

For example, f(x+1)=-f(3+x), find the period of f(x).

1. Make a variable substitution so y=x+1 to get f(y)= f(y+2);

2. Apply this formula again to get f(y+2)=-f(y+4);

3. The two formulas are combined to obtain f(y)=f(y+4), so the period is 4.

The key point is: change the calendar to make up f(x)=f(x+t), and at this time t is the cycle. And the above 3 steps are to search in this direction.

The formula for period t is Wang Xiao:1. T 2 r v (linear velocity of the circumference of the periodic circle).

2, t 2 for angular velocity).

The essence of the periodic function: when the difference between the values of the two independent variable values is equal to the multiple of the period, the function values of the two independent variable values as a whole are equal. For example, f(x+6) =f(x-2), then the period of the function is t=8.

Periodic function properties:

1) If t(≠0) is the period of f(x), then -t is also the period of f(x).

2) If the source type t(≠0) is the period of f(x), then nt(n is any non-zero integer) is also the period of f(x).

3) If t1 and t2 are both periods of f(x), then t1 t2 is also periods of f(x).

4) If f(x) has a minimum positive period t*, then any positive period t of f(x) must be a positive integer multiple of t*.

5) The domain m of the periodic function f(x) must be an unbounded hail guess set of both sides.

How to find the function period: y=sinx cosx=tanx, t=pi.

The mathematics is described as follows:

mathematics [English: mathematics, from the ancient Greek máthēma); Often abbreviated as math or maths], it is a discipline that studies concepts such as quantity, structure, change, space, and information.

Mathematics is a general means for human beings to strictly describe and deduce the abstract structure and pattern of things, which can be applied to any problem in the real world, and all mathematical objects are artificially defined in nature. In this sense, mathematics belongs to the formal sciences, not the natural sciences. Different mathematicians and philosophers have a range of opinions on the exact scope and definition of mathematics.

In the historical development and social life of mankind, mathematics plays an irreplaceable role, and it is also an indispensable basic tool for learning and researching modern science and technology.

Mathematics originated in the early production activities of human beings, and the ancient Babylonians had accumulated a certain amount of mathematical knowledge since ancient times and could apply practical problems. From the perspective of mathematics itself, their mathematical knowledge is only obtained through observation and experience, and there is no comprehensive conclusion and proof, but it is also necessary to fully affirm their contributions to mathematics.

The knowledge and application of basic mathematics is an integral part of individual and group life. The refinement of its basic concepts can be seen in ancient mathematical texts in ancient Egypt, Mesopotamia, and ancient India. Since then, there has been a steady stream of development.

But algebra and geometry at that time remained independent for a long time.

Algebra is arguably the most widely accepted form of "mathematics". It can be said that since everyone started learning to count when they were young, the first mathematics they came into contact with was algebra. As a discipline that studies "numbers", algebra is also one of the most important components of mathematics.

Geometry was the first branch of mathematics to be studied.

It wasn't until the Renaissance in the 16th century that Descartes founded Zen analytic geometry, linking algebra and geometry, which were completely separate at the time. Since then, we can finally prove the theorems of geometry with calculations; At the same time, it can also be used to visualize abstract algebraic equations and trigonometric functions. Later, calculus was developed into more subtle forms.

There are only three derivations of the periodicity of the function, which are as follows:

1. If the function f(x)(x d) has two axes of symmetry x=a and x=b in the defined noisy state domain, then the function f(x) is a periodic function, and the period t=2|b-a|(Not necessarily a minimum positive period).

2. If the function f(x)(x d) has two symmetry centers a(a,0) and b(b,0) in the defined domain, then the function f(x) is a periodic function, and the period t=2|b-a|(Not necessarily a minimum positive period).

3. If the function f(x)(x d) has an axis of symmetry x=a and a pair of axes that happen to be called the center b(b, 0)(a≠b) in the defined domain, then the function f(x) is a periodic function, and the period t=4|b-a|(Not necessarily a minimum positive period).

The periodic function properties are as follows:

1) If t(≠0) is the period of f(x), then -t is also the period of f(x).

2) If t(≠0) is a period of f(x), then nt(n is an arbitrary non-zero integer) is also a period of f(x).

3) If t1 and t2 are both periods of f(x), then t1 t2 is also periods of f(x).

4) If f(x) has a minimum positive period t*, then any positive period t of f(x) must be a positive integer multiple of t*.

y=sinx, the ordinate remains unchanged, after the abscissa becomes twice as much, it is called that y=abscissa does not change, and after the ordinate becomes twice as much, it is y=2sinx, for y=sinwx

Cycle t is 2 W

If the ordinate does not change, the abscissa becomes the original T times.

From the image, we can see that the period t'=2 t w=2 (w t), that is, Zheng Ji w'=w t

To find the period, you can subdivide a function into the form of f(x)=f(x+a), then its period is a (of course a 0), for example, the following is a series of 2a as a periodic functions.

f(x+a)=-f(x) So there is f(x+a+a)=-f(x+a)=f(x) to dissolve into the form of f(x)=f(x+2a), the key is to use the overall idea to substitute.

Definition of the periodic state residuality of the function: If there is a constant t, for any x in the defined domain, so that f(x)=f(x+t) is constant, then f(x) is called the periodic function, and t is called a period of this function.

To find the period, you can subdivide a function into the form of f(x)=f(x+a), then its period is a (of course a 0), for example, the following is a series of 2a as a periodic functions.

f(x+a)=-f(x) So there is f(x+a+a)=-f(x+a)=f(x) to dissolve into the form of f(x)=f(x+2a), the key is to use the overall idea to substitute.

Definition of the periodic state residuality of the function: If there is a constant t, for any x in the defined domain, so that f(x)=f(x+t) is constant, then f(x) is called the periodic function, and t is called a period of this function.

To find the period, you can subdivide a function into the form of f(x)=f(x+a), then its period is a (of course a 0), for example, the following is a series of 2a as a periodic functions.

f(x+a)=-f(x) So there is f(x+a+a)=-f(x+a)=f(x) to dissolve into the form of f(x)=f(x+2a), the key is to use the overall idea to substitute.

Periodic definition of a function: If there is a constant t, for any x in the defined domain, so that f(x)=f(x+t) is constant, then f(x) is called the periodic function, and t is called a period of the function.