Concepts and properties of functions, concepts and properties of functions

On [0,7], only f(1)=f(3)=0f(5)≠0, and f(2-x)=f(2+x)f(-1)=f(5)≠0

f(-1)≠f(1)=0

f(-1)≠±f(1)

That is, the function f(x) is neither odd nor even.

f(2-x)=f(2+x), f(x)=f(4-x);

f(7-x) = f(7+x), f(x) = f(14-x);

So f(4-x) = f(14-x).

f(4-(4-x))) = f(14-(4-x)) gives f(x) = f(x+10).

f(x) is a periodic function with a minimum positive period of 10

When n is an integer, f(10n+1)=f(1)=0, f(10n+3)=f(3)=0, where -2005 10n+1 2005, -2005 10n+3 2005,, these two inequalities have 401 integer solutions respectively, that is, the equation f(x)=0 has 802 roots.

The function f(x) is over the closed interval [0,7], and only f(1) f(3) 0

f(5)≠0

f(2 x) f(2+x), f(1) f(5), f(1) ≠0, f(1) 0

f( 1) ≠ f(1), the function f(x) is neither odd nor even.

f(2－x)＝f(2+x)，→f(4－x)＝f(x)f(7－x)＝f(7+x)，→f(4－x)＝f(10+x)f(x)＝f(10+x)

10 is a period of the function f(x).

f(7 x) f(7+x), the function f(x) has no root on [4,7].

The function f(x) has no root on [7,10].

f(x) 0 has exactly two roots of 1 and 3 on [0,10], and the roots of f(x) 0 are of the form 10n+1 or 10n+3.

2005 10n+1 2005, 200 n 200, a total of 201.

2005 10n+3 2005, 200 n 200, a total of 201.

The number of roots of equation f(x) 0 in the closed interval [2005,2005] is 802

Eraser, eraser, eraser

Someone replied that it was faster than me and better than me...

The concept and properties of functions are as follows:

1. The popular meaning of a function is that it consists of independent variables.

There may be one, two, or n independent variables, but the value of the dependent variable is also uniquely determined when the independent variable is determined.

2. The meaning of function is that in the field of mathematics, a function is a relation, which makes each element in one set correspond to the only element in another set.

The nature of the function

1. Boundedness.

Let the function f(x) be defined in the interval x, and if m>0 exists, there is always a | for all x on the interval xf（x）|m, then f(x) is said to be bounded in the interval x, otherwise f(x) is said to be unbounded in the interval.

2. Monotonicity.

Let the domain of the function f(x) be defined.

is d, and interval i is contained in d. If for any two points x1 and x2 on the interval, when x1 is called the function f(x) is monotonically increasing on the interval i; If for any two points x1 and x2 on interval i, when x1f(x2), then the function f(x) is said to be monotonically decreasing on interval i. Monotonically increasing and monotonically decreasing functions are collectively referred to as monotonic functions.

Function concept: Let a and b be non-empty sets of numbers, and if, according to some definite correspondence f, there is a uniquely definite number f(x) corresponding to any number x in set a, then f:a b is called a function from set a to set b.

Quality. Property 1: Symmetry.

Number axis symmetry: The so-called number axis symmetry means that the function image is symmetrical with respect to the axes x and y.

Origin symmetry: Again, such symmetry means that the coordinates of the coordinates of the points on the function of the image symmetry with respect to the origin, on both sides of the origin, are opposite to each other.

About point symmetry: This type is quite similar to origin symmetry, except that the symmetry point is no longer limited to the origin, but any point on the coordinate axis.

Nature 2: Periodicity.

The so-called periodicity means that the image of the function in a part of the region is repeated, assuming that a function f(x) is a periodic function, then there is a real number t, when x in the defined field is added or subtracted by an integer multiple of t, the y corresponding to x does not change, then it can be said that t is the period of the function, if the absolute value of t reaches the minimum, it is called the minimum period.