Help summarize all the properties of functions, the basic properties of functions

Its properties usually refer to the definition domain, value range, analytical, monotonicity, parity, periodicity, and symmetry of the function. A function represents a correspondence in which each input value corresponds to a unique output value. The standard symbol for the output value corresponding to the input value x in the function f is f(x).

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.

The properties of the function are:

Conclusions about the tonality of the monochromatic type

1. If f(x) and g(x) are both increase (decrease) functions, then f(x)+g(x) are still increase (decrease) functions.

2. Two functions that are inverse functions of each other have the same monotonicity.

3. y=f[g(x)] is a function defined on m, if f(x) and g(x) have the same monotonicity, then its composite function f[g(x)] is the number of increasing areas; If the monotonicity of f(x) and g(x) is opposite, then the composite function f[g(x)] is a subtraction function, referred to as "same increase and different subtraction".

4. The monotonicity of the odd function is the same in the two intervals with respect to the symmetry of the original comma point; The even function has opposite monotonicity on the two-virtual guess interval with respect to the origin symmetry.

Conclusions about parity

1. The symmetrical nature of the image: the sufficient and necessary condition for a function to be an odd function is that its image is symmetrical with respect to the origin, and the sufficient and necessary condition for a function to be an even function is that its image is symmetrical with respect to the y-axis.

2. Let the definition domains of f(x) and g(x) be d1 and d2 respectively, then on their common definition domains: odd + odd = odd, odd x odd = even, even + even = even, even x even = even, odd x even = odd.

3. The function f(x) of any domain with respect to the origin symmetry can be written as an odd function g(x) and an even function h(x) and form.

Important conclusions of the cyclical nature

1. f(x+a)=f(x), then y=f(x) is a periodic function with t=a as the period;

2. If the function y=f(x) satisfies f(x+a)=-f(x)(a>0), then f(x) is a periodic function and 2a is one of its periods.

3. If the function f(x+a)=f(x-a), it is a periodic function with t=2a as the period.

4. y=f(x) satisfies f(x+a)=1 f(x)(a>0), then f(x) is a periodic function and 2a is one of its periods.

If the function y=f(x) satisfies f(x+a)=-1 f(x)(a>0), then f(x) is a periodic function and 2a is one of its periods.

The basic properties of functions include boundedness, monotonicity, parity, and continuity. Let it be a real function of a real variable, if there is f(-x)=-f(x), then f(x) is an odd function. Let f(x) be a real function of a real variable, and if f(x)=f(-x), then f(x) is an even function.

Continuity is a property of a function, and a continuous function is a function in which when the change in the input value is small enough, the change in the output will be small enough. The basic properties of functions include boundedness, monotonicity, parity, and continuity. Let it be a real function of a real variable, if there is f(-x)=-f(x), then f(x) is an odd function.

Let f(x) be a real function of a real variable, and if f(x)=f(-x), then f(x) is an even function. Continuity is functional'A continuous function is a function in which the change in the input value is small enough, and the change in the output is small enough.