Mathematical analysis divides basic elementary functions into six categories: power function, exponential function, logarithmic function, trigonometric function, inverse trigonometric function and constant function.
These functions are described below.
1, power function
definition
Generally speaking, a function in the form of y=xα(α is a rational number), that is, a function with the base as the independent variable, the power as the dependent variable and the exponent as the constant is called a power function. For example, functions y=x0, y=x 1, y=x2, y=x- 1 (note: x = x-1=1x = x0 when x≠0) are all power functions. The general form is as follows:
(α is a constant, which can be a natural number, a rational number, or any real number or complex number. )
2. Definition of exponential function
Exponential function is an important function in mathematics. This function applied to the value e is written as exp(x). It can also be written as ex, where e is a mathematical constant and the base of natural logarithm, which is about equal to 2.7 1828 1828, also known as Euler number. The general form is as follows:
(a & gt0，a≠ 1)
3. Logarithmic function
definition
In general, the function y = logax(a >；; 0, and a≠ 1) is called logarithmic function, that is, a function with power (real number) as independent variable, exponent as dependent variable and base constant as constant is called logarithmic function.
Where x is the independent variable and the domain of the function is (0, +∞), that is, x >；; 0。 It is actually the inverse function of exponential function, which can be expressed as x=ay. Therefore, the stipulation of a in exponential function is also applicable to logarithmic function. The general form is as follows:
(a>0, a≠ 1, x>0, especially when α=e, denoted as y=ln x)
4, trigonometric function
Trigonometric function is a common function about angle in mathematics. That is to say, a function with angle as the independent variable and the ratio of angle to any two sides as the dependent variable is called trigonometric function, which relates the internal angle of a right triangle to the ratio of the lengths of its two sides, and can also be equivalently defined as the lengths of various line segments related to the unit circle.
Trigonometric function plays an important role in studying the properties of geometric shapes such as triangles and circles, and is also a basic mathematical tool for studying periodic phenomena. In mathematical analysis, trigonometric function is also defined as infinite limit or the solution of a specific differential equation, which allows its value to be extended to any real value or even complex value.
5. Inverse trigonometric function
Inverse trigonometric function is a basic elementary function. It is the general name of the functions of arcsine x, arccosine arccos x, arctangent, arctangent x, arctangent x, arctangent x, arctangent x, arctangent x, and arctangent x, which respectively represent their arcsine, arccosine, arctangent and cotangent. Tangent and cotangent are the angles of X.
It can't be understood as the inverse function of trigonometric function in a narrow sense, but it is a multi-valued function. The inverse function of trigonometric function is not a single-valued function, because it does not meet the requirement that the independent variable corresponds to a function value, and its image is symmetrical with its original function about function y = X. Euler put forward the concept of inverse trigonometric function, and expressed it in the form of "arc+function name" for the first time.
6. Constant function
definition
In mathematics, a constant function (also called a constant function) refers to a function whose value is constant (that is, constant). For example, we have the function f(x)=4, because f maps any value to 4, so f is a constant. More generally, for a function f: A→B, if f(x)=f(y) exists for all x and y in a, then f is a constant function.
Please note that it is meaningless for every empty function (a function whose domain is an empty set) to meet the above definition, because there is no X and Y in A, which makes f(x) and f(y) different. However, some people think that constant functions will be easier to define if they contain empty functions.
For polynomial functions, non-zero constant functions are called zeroth degree polynomials. The following is the general form:
Y=C (C is a constant)
Extended data
Characteristics of functions
1, Boundedness
Let the function f(x) be defined on the interval x, if there is m >；; 0, for all X belonging to the interval X, there is always |f(x)|≤M, then f(x) is bounded on the interval X, otherwise F (x) is unbounded on the interval.
2. Monotonicity
Let the domain of the function f(x) be D, and the interval I is contained in D. If for any two points x 1 and x2 on the interval, when X 1
3. Parity
set up
Real function is a real variable. If f(-x)=-f(x), then f(x) is odd function. Geometrically, a odd function is symmetrical about the origin, that is, its image will not change after it rotates 180 degrees around the origin. Odd function's examples are X, sin(x), sinh(x) and erf(x). Let f(x) be a real function, if any.
Then f(x) is an even function.
Geometrically, an even function is symmetrical about the Y axis, that is, its graph will not change after being mapped to the Y axis. Examples of even functions are |x|, x2, cos(x) and cosh(x). Even functions cannot be bijective mappings.
4. periodicity
Let the domain of function f(x) be d. If there is a positive number t, then for any one
have
And f(x+T)=f(x) is a constant, then f(x) is called a periodic function and t is called the period of f(x). Usually we say that the period of a periodic function refers to the minimum positive period. The domain D of a periodic function is an unbounded interval with at least one side. If d is bounded, the function is not periodic. Not every periodic function has a minimum positive period, such as Dirichlet function.
Periodic functions have the following properties:
(1) If T(T≠0) is the period of f(x), then -T is also the period of f(x).
(2) If T(T≠0) is the period of f(x), then nT(n is an arbitrary non-zero integer) is also the period of f(x).
(3) If T 1 and T2 are both periods of f(x), then
It is also the period 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)T* is the minimum positive period of f(x), and T 1 and T2 are two periods of f(x) respectively, then T 1/T2∈Q(Q is a rational number set).
(6) If T 1 and T2 are two periods of f(x) and T 1/T2 is an irrational number, then f(x) does not have a minimum positive period.
(7) The domain m of the periodic function f(x) must be a set with unbounded sides.
5. Continuity
In mathematics, continuity is an attribute of a function. Intuitively, a continuous function is a function in which the change of the input value is small enough and the change of the output is small enough. If a small change in the input value will cause a sudden jump, or even the output value is uncertain, the function is called a discontinuous function (or discontinuous function).
Let f be a function of the projection of a subset of a real number set: f is continuous at point C if and only if the following two conditions are met:
F is defined at point C, and C is one of the polymerization points. No matter how the independent variable X approaches C in the equation, the limit of f(x) exists and is equal to f(c). We say that a function is continuous everywhere or everywhere, or if it is continuous at any point in its definition domain, it is simply continuous. More generally, we say that a function is continuous on a subset of its domain, when it is continuous at every point on this subset.
Without the concept of limit, the continuity of real function can also be defined by the following so-called method.
Let's consider the function. Suppose c is an element in the domain of f. The function f is continuous at point C if and only if the following conditions are true:
For any positive real number, there is a positive real number δ >; 0 so that for δ in any domain, as long as x satisfies c-? δ& lt； X & ltC+δ, established.
Baidu Encyclopedia-Basic Elementary Function
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