1. Function
(1) Definition of function : If when the variable x is arbitrarily set to a value within its variation range, the quantity y always has a certain value corresponding to it according to a certain rule f, then y is called a function of x . The range of the variable x is called the domain of the function . Usually x is called the independent variable , y is called the function value (or dependent variable ) , and the variation range of the variable y is called the range of the function . Note : To show that y is a function of x, we use the notation y=f(x), y=F(x), and so on. The letters "f" and "F" here represent the corresponding law between y and x, that is, the functional relationship, and they can be represented by different letters arbitrarily. If the independent variable takes a definite value in the definition domain, the function has only one definite value corresponding to it, this kind of function is called a single-valued function, otherwise it is called a multi-valued function. Here we only discuss single-valued functions.
⑵, the function is equal
From the definition of the function, we can know that the constituent elements of a function are: the domain of definition, the corresponding relationship and the range of values. Since the value domain is determined by the domain of definition and the corresponding relationship, if the domain of definition and the corresponding relationship of the two functions are exactly the same, we say that the two functions are equal .
(3) Representation method of domain function
a): Analytical method: The method of expressing the correspondence between the independent variable and the dependent variable with a mathematical formula is the analytical method. Example: In the Cartesian coordinate system, the equation of a circle with radius r and center at the origin is: x 2 +y 2 =r 2
b): Table method: The method of listing a series of independent variable values and corresponding function values in a table to represent the functional relationship is the table method. Example: In practical applications, the square tables and trigonometric function tables that we often use are all functions expressed in tables.
c): Graphical method: The method of representing a function with a curve on the coordinate plane is the graphic method. Generally, the abscissa is used to represent the independent variable, and the ordinate is used to represent the dependent variable. Example: In the Cartesian coordinate system, a circle with a radius r and a center at the origin is represented graphically as:
2, the simple nature of the function
(1) The boundedness of the function : if all x values belonging to a certain interval I always have │f(x)│≤M, where M is a constant independent of x, then we call f(x) in The interval I is bounded, otherwise it is called unbounded.
Note: A function is called a bounded function if it is bounded in its entire domain
( 2 ) Monotonicity of the function : If the function increases as x increases in the interval (a, b), that is: for any two points x 1 and x 2 in (a, b) , when x 1 < x 2 When , there is , the function is said to be monotonically increasing in the interval (a, b) . If the function decreases with the increase of x in the interval (a, b), that is: for any two points x 1 and x 2 in (a, b) , when x 1 < x 2 , there is, then it is called a function It is monotonically decreasing in the interval (a,b) .
(3) The parity of the function
If the function satisfies = for any x in the domain , it is called an even function; if the function satisfies =- for any x in the domain , it is called an odd function.
Note: The graph of the even function is symmetric about the y-axis, and the graph of the odd function is symmetric about the origin.
⑷, the periodicity of the function
For a function, if there is a non-zero number l such that the relation is true for any value of x in the domain, it is called a periodic function , where l is the period.
Note: The period of the periodic function we say refers to the minimum positive period.
3. Inverse function
(1) Definition of inverse function : If there is a function , if the variable y takes any value y 0 in the value domain of the function , the variable x must have a value x 0 corresponding to it in the definition domain of the function, that is , then the variable x is a variable A function of y. This function is used to represent and is called the inverse of a function .
Note: From this definition, a function is also the inverse of a function.
( 2 ) The existence theorem of the inverse function : If (a, b) is strictly increased (decreased), and its value range is R , then its inverse function must be determined on R , and it is strictly increased (decreased).
Note: Strict increase (decrease) is monotonous increase (decrease)
(3) The nature of the inverse function : in the same coordinate plane, the graph of and is symmetrical about the straight line y=x.
4. Compound functions
Definition of composite function : If y is a function of u: , and u is a function of x: , and all or part of the function value of y is in the domain of definition, then the connection of y through u is also a function of x, we call it after A function is a function composed of a function and a compound, referred to as a compound function, denoted as , where u is called an intermediate variable.
Note: Not any two functions can be composited; composite functions can also be composed of more functions.
5. Elementary functions
(1) Basic elementary functions : There are five basic elementary functions that we most commonly use, namely: exponential function, logarithmic function, power function, trigonometric function and inverse trigonometric function. Below we summarize them in a table:
| function name |
function notation |
graph of function |
properties of functions |
| Exponential function |
a): no matter what the value of x is, y is always positive; |
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| Logarithmic function |
a): The graph is always located on the right side of the y-axis and passes the (1,0) point |
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| Power function |
a is any real number |
|
Let a=m/n |
| Trigonometric functions |
(Sine function) |
a): The sine function is a periodic function with a period of 2π |
|
| Inverse trigonometric functions |
(Arcsine function) |
a): Since this function is a multi-valued function, we limit the value of this function to [-π/2, π/2], and call it the principal value of the arcsine function. |
2. Elementary function : A function that is produced by a finite number of rational operations and a finite number of function combinations with a basic elementary function and a constant and can be expressed by an analytical expression is called an elementary function.
6. Hyperbolic function and inverse hyperbolic function
(1) Hyperbolic function : The hyperbolic function we often encounter in applications is: (described in a table)
| the name of the function |
function expression |
graph of function |
properties of functions |
| Hyperbolic sine |
a): Its domain of definition is: (-∞,+∞); |
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| Hyperbolic cosine |
a): Its domain of definition is: (-∞,+∞); |
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| hyperbolic tangent |
a): Its domain of definition is: (-∞,+∞); |
Let's take a look at the difference between hyperbolic functions and trigonometric functions:
| Properties of Hyperbolic Functions |
Properties of Trigonometric Functions |
| shx and thx are odd functions, chx is even function |
sinx and tanx are odd functions, cosx is an even function |
| Neither of them are periodic functions |
are periodic functions |
Hyperbolic functions also have sum and difference formulas:
(2) Inverse hyperbolic function : The inverse function of a hyperbolic function is called an inverse hyperbolic function .
a)：反双曲正弦函数 其定义域为：(-∞,+∞)；
b)：反双曲余弦函数 其定义域为：[1,+∞)；
c)：反双曲正切函数 其定义域为：(-1,+1)；