Directory, introduction, complex function, inverse function, implicit function, multivariate function, quadratic function, linear function, trigonometric function, brief introduction to the development history of function concepts. In the field of mathematics, a function is a relationship that makes each element in one set correspond to the only element in another (possibly the same) set. This is only the case of unary function f (x) = y, please give a general definition according to the original English text, thank you. - A？ Variable? So what? Related? Where to? Another one? That? For what? Every one? Value? Hypothetically? By who? One? There? Is it? Answer? Value? Make up your mind? For what? That? Others. A variable whose function is related to another quantity. Any value in this quantity can find a corresponding fixed value in another quantity. - A？ Rules? Yes? Communication? Between? Two? Set? Like this? That? There? Is it? Answer? Unique? Elements? Are you online? That? Second? Settings? Distribution? Where to? Every one? Elements? Are you online? That? First of all? Settings. The rule of one-to-one correspondence between two groups of elements of a function. Each element in the first group has only a unique corresponding number in the second group. The concept of function is the most basic for every branch of mathematics and quantity. Function? Correspondence in mathematics is the correspondence from real number set A to real number set B. Simply put, A varies with B, and A is a function of B? . To be precise, let x be a non-empty set and y be a real set. F is a rule? ,? If for each X in X, according to rule F, there is a Y corresponding to it? ,? We call f a function on x, and we call it y = f(x). Let's call x the domain of function f (x), y the range, x the independent variable and y the dependent variable. ? Example 1: y = sinx? X=[0，2π]，Y=[- 1， 1]？ This gives a functional relationship. Are you sure? Change y to y 1 = (a, b)? , a < b is an arbitrary real number or a functional relationship. ? What is its depth y and shore point? O Distance to the measuring point? x？ The corresponding relationship between them is curvilinear, representing a function with the domain [0, b]. The above three examples show three representations of functions: formula method? ,? Table method and image method. ? A compound function has three variables, y is the function of u, y = ψ (u), u is the function of x, and u = f (x), which can often form a chain: y forms the function of x through the intermediate variable u:? X→u→y, depending on the domain: let the domain of ψ be u? . ? The range of f is u when U*? U, called f and ψ? Form a composite function? ,? Like what? Y=lgsinx, x∈(0, π). At this time, Sinx > 0? , lgsinx makes sense? . But if x ∈ (-π, 0) is specified, then sinx < 0? Lgsinx doesn't make sense? , it cannot be a composite function. ? Inverse functions are generally two-way in relation? So is the function? Let y = f (x) be a known function. If there is a unique x∈X for every y∈Y, so that f (x) = y, is it the process of finding x from y? So x becomes a function of y? , recorded as x = f? -1(y). It's called F? -1 is the inverse function of f, and it is customary to use x as the independent variable. So this function is still written as y = f? - 1(x)？ , for example? Y = sinx and y = arcsinx? Inverse function of each other. In the same coordinate system, y = f (x) and y = f? Is the graph of-1(x) symmetric about the straight line y = x? What if the implicit function can be derived from the functional equation? F(x，y)=0？ Determine that y is the function of x y = F(x), that is, F(x, f(x))≡0, which is called the implicit function of x? Set point of multivariate function (x 1, x2, …, xn)? ∈G？ Rn，U？ R 1？ If there is a unique rule f for each point (x 1, x2, …, xn)∈G? U∈U corresponds to: f: g → u, u = f (x 1, x2, …, xn), then f is called a function of n variables, g is the domain, and u is the range. ? Basic elementary functions and their images? Power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function are called basic elementary functions. ? ① Power function: y = x μ (μ ≠ 0, μ is any real number) Definition domain: μ is a positive integer: (-∞, +∞), μ is a negative integer: (-∞, 0)∞(0, +∞); μ = (α is an integer), and when α is an odd number, it is (? -∞, +∞), (0, +∞) When α is even; μ = p/q, p, q is coprime, and is a composite function of. Sketches are shown in Figures 2 and 3. ? ② exponential function: y = ax (a > 0? , a≠ 1), defined as (? -∞, +∞), the range is (0? ，+∞)，a>0？ Time is a strictly monotonically increasing function (? That is, when x2 > x 1 ，0 0)，？ Call a bottom? ,? The domain is (0, +∞) and the range is (-∞, +∞)? . a> 1？ Time is strictly monotonically increasing, and it is strictly monotonically decreasing when 0 < a < 1 No matter what the value of a is, the graph of logarithmic function passes through the point (1, 0), and both logarithmic function and exponential function are reciprocal functions. . As shown in fig. 5. ? The logarithm with the base of 10 is called ordinary logarithm? Jane is lgx? . Logarithm based on e, that is, natural logarithm, is widely used in science and technology, and is recorded as lnx. ? ④ Trigonometric function: See Table 2. ? Sine function and cosine function are shown in figs. 6 and 7. ? ⑤ Inverse trigonometric function: See Table 3. Hyperbolic sine and cosine are shown in figure 8. ? ⑥ Hyperbolic function: hyperbolic sine (ex-e-x), hyperbolic cosine? (ex+e-x), hyperbolic tangent (ex-e-x)/(ex+e-x)? , hyperbolic cotangent (? ex+e-x)/(ex-e-x).？ [Editor] Supplement In the field of mathematics, a function is a relationship, which makes each element in one set correspond to the only element in another (possibly the same) set (this is only the case of unary function f (x) = y, please give a general definition according to the original English text, thank you). The concept of function is the most basic for every branch of mathematics and quantity. ? The terms function, mapping, correspondence and transformation usually have the same meaning. ?

3.？ Note the following attributes:

Know its origin: If B is a subset of A, then the element a 1 has two choices (in or out). Similarly, for element a2,? A3, ... Ann, there are two choices, so there is always * * *? What kind of choice? That is, set a has it? A subset.

Of course, we should also pay attention to this? In this case, it includes the case that no N elements exist, so what is the number of proper subset? What is the number of nonempty proper subset?

(3) De Morgan's Law:

Some versions may be written like this, and you should be able to understand them after encountering them.

4.？ Will you solve the problem with the idea of complement set? (Exclusion method, indirect method)

The value range of.

Note that sometimes you can get a lot of information from the set itself, so don't miss it when you do the problem; ? If I tell you that the function F (x) = AX2+BX+C (A > 0)? Are you online? Monotone decreasing, in? Monotonically increasing on, you should immediately know that the symmetry axis of the function is x= 1. Or am I right? Should also immediately think that m and n are actually equations? The two roots of ""

5, familiar with several forms of propositions,

What are the four forms of proposition and their relationships?

A proposition with reciprocal negation is an equivalent proposition. )

Both the original proposition and the negative proposition are true and false; Whether it is an inverse proposition or not, a proposition is the same as true or false.

6, familiar with the nature of the necessary and sufficient conditions (college entrance examination)

Meet the requirements? ,? Meet the requirements? ,

What if? ; then what what's up A sufficient and unnecessary condition of? ;

What if? ; then what what's up Necessary but not sufficient conditions for? ;

What if? ; then what what's up Necessary and sufficient conditions? ;

What if? ; then what what's up It is neither sufficient nor necessary. ;

7.？ Do you know the concept of mapping? Mapping F: A → B, have you noticed the arbitrariness of elements in A and the uniqueness of corresponding elements in B? What kind of correspondence can form a mapping?

(one-to-one, many-to-one, allowing the elements in B to have no original image. )

Pay attention to the solution of mapping number. If there are m elements in set A and n elements in set B, the number of mappings from A to B is nm.

Such as: if? ,? ; Q:? Arrive? There is one. Arrive? What is the mapping of? a； ? Arrive? What is the function of? First, if? And then what? Arrive? What is a one-to-one mapping? A.

Function? Images and lines? What is the number of intersections? A.

8.？ What are the three elements of a function? How to compare whether two functions are the same?

(Definition domain, corresponding rule, value domain)

The judgment method of the same function: ① the expressions are the same; (2) Domain consistency? (These two points must be met at the same time)

9.？ What are the common types of finding function domain?

Solution of function definition domain;

The denominator in the score is not zero;

The number (or formula) under even roots is greater than or equal to zero;

Exponential radix is greater than zero and not equal to one;

The base of logarithmic formula is greater than zero, which is not equal to one, and the true number is greater than zero.

Tangent function

Cotangent function

Definition domain of inverse trigonometric function

What is the domain of the function y = arcsinx? [- 1,? 1] ? What is the scope? What is the domain of the function y = arccosx? [- 1,? 1]? What is the scope? [0,? π]? What is the domain of the function y = arctgx? r？ What is the scope? . What is the domain of the function y = arcctgx? r？ What is the scope? (0,? π)? .

When the above two or more aspects appear at the same time, first find out the range of independent variables satisfying each condition, and then take their intersection to get the definition domain of the function.

10.？ How to find the domain of compound function?

The meaning domain is _ _ _ _ _ _ _ _. ?

power function

concept

A function in the form of y = x a (a is a constant), that is, taking the base as the independent variable? A function whose power is a dependent variable and whose exponent is a constant is called a power function. A is easy to understand when it takes a nonzero rational number, but it is not easy for beginners to understand when A takes an irrational number. Therefore, in the elementary function, we don't need to master the problem that the exponent is irrational, we just need to accept it as a known fact, because it involves the profound knowledge of the real number continuum.

characteristic

For a nonzero rational number, it is necessary to discuss their respective characteristics in several cases: firstly, we know that if a=p/q, and p/q is an irreducible fraction (that is, p and q are coprime), and both q and p are integers, then x (p/q) = the power of q (the power of x), if q is odd, then the domain of the function is r, and when the exponent. One is that it may be used as the denominator but not 0, and the other is that it may be under even root but not negative. Then we can know:

When a is less than 0, x is not equal to 0; ?

When q is an even number, x is not less than 0; ?

When q is odd, x takes r.

Definition domain and value domain

When a is different, the different situations of the power function domain are as follows:

1. If a is negative, then X must not be 0, but at this time the domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the domain of the function is all real numbers greater than 0;

2. If q is an odd number at the same time, the domain of the function is not equal to 0? All real numbers of.

When x is different, the range of power function is different as follows:?

1. When x is greater than 0, the range of the function is always a real number greater than 0.

2. When x is less than 0, then only at the same time Q is odd, and the range of the function is non-zero real number. Only when a is a positive number will 0 enter the value range of the function. Since x is greater than 0, it is meaningful to any value of a, so the following gives the respective situations of power function in the first quadrant.

Particularity of the first quadrant

Can you have a look?

(1) All graphics pass (1, 1). (a≠0)？ A> at 0 o'clock? Image intersections (0,0) and (1, 1)?

(2) When a is greater than 0, the power function monotonically increases to increasing function? When a is less than 0, the power function monotonically decreases to a decreasing function. ?

(3) When a is greater than 1, the power function graph is convex downward (vertically thrown); When a is less than 1 greater than 0, the power function graph is convex (horizontally thrown). When a is less than 0, the image is hyperbolic. ?

(4) When a is less than 0, the smaller A is, the greater the inclination of the graph is. ?

(5) Obviously, the power function has no boundary. ?

(6)a=2n, and the function is even? {x|x≠0} .

picture

Power function image:? ① When a≤- 1 and a is odd, is the function a decreasing function in the first and third quadrants? ② when a≤- 1 and a is even, the function is increasing function in the second quadrant and the function is subtraction in the first quadrant? ③ When a=0 and x is not 0, the function image is parallel to the x axis, and y= 1 instead of (0, 1)? ④ When 0