1. function: Let A and B be non-empty sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B, denoted as y=f(x) and x ∈.
2. Thinking about solving problems in function definition domain;
(1) If x is in the denominator position, the denominator x cannot be 0.
⑵ The number of even roots is not less than 0.
(3) The truth value of the logarithmic formula must be greater than 0.
(4) The base of exponential logarithm cannot be 1 and must be greater than 0.
5] When the index is 0, the cardinal number shall not be 0.
[6] If a function is a combination of some basic functions through four operations, then its domain is the set of all meaningful X values.
Once the domain of the function in the actual problem is defined, it is necessary to ensure that the actual problem is meaningful.
3, the function is the same
⑴ The expression is the same: it has nothing to do with the letters representing independent variables and function values.
⑵ The definition fields are consistent and the corresponding laws are consistent.
4. The solution of function value domain
⑴ Observation method: it is suitable for elementary functions and some simple functions obtained by four operations of elementary functions.
⑵ Image method: It is suitable for functions that are easy to draw function images.
⑶ Matching method: mainly used for quadratic function, and the formula is y = (x-a) 2+b.
⑷ Replacement method: it is mainly used to infer the range of unknown functions from functions with known range.
5. Transformation of function image
(1) Translation transformation: the transformation on the X axis is added and subtracted on X, and the transformation on the Y axis is added and subtracted on Y. ..
⑵ Telescopic transformation: add coefficient before X. ..
⑶ Symmetric transformation: no requirement for senior high school.
6. Mapping: Let A and B be two non-empty sets. If the element X of any tool in A has a uniquely determined Y corresponding to it in set B according to a corresponding rule F, then the corresponding F: A → B is called the mapping from set A to set B. ..
(1) Every element in set A has an image in set B, and this image is unique.
⑵ Different elements in set A, the corresponding images in set B can be the same.
⑶ Every element in set B is not required to have an original image in set A. ..
7, piecewise function
⑴ There are different analytical expressions in different parts of the domain.
⑵ The range of independent variables and function values of each part is different.
(3) The domain of piecewise function is the intersection of the domain of each segment, and the range is the union of the range of each segment.
8. Compound function: If (u∈M) and u=g(x) (x∈A), then y=f[g(x)]=F(x) (x∈A), which is called the compound function of f and g.
2 the nature of the mathematical function of senior one
1, the local property of the function-monotonicity
Let the domain of function y=f(x) be I, if any two variables x 1 in an interval d in the domain I correspond to x2, when X 1
(1) Judgment of monotonicity of function interval
I if x 1 and x2 are randomly selected in a given interval, then x 1 and x2∈D, and x 1
Ⅱ. Do the difference f(x 1)-f(x2), and do the deformation and formula, which is convenient for judging the positive and negative.
Ⅲ. Judge the sign of deformation expression f(x 1)-f(x2) and point out monotonicity.
⑵ Monotonicity of compound function
The monotonicity of the compound function y=f[g(x)] is closely related to the monotonicity of its constituent functions u=g(x) and y=f(u), and its law is "same increase but different decrease"; The compound function of multivariate function is based on the principle of "decreasing even number will increase, and decreasing odd number will decrease".
(3) Precautions
The monotone interval of a function can only be a subinterval of its domain, and the intervals with the same monotonicity cannot be summed together to write a union. If the function increases in both intervals A and B, the monotonically increasing interval expressed as f(x) is A and B, and cannot be expressed as A ∪ B..
2. The global nature of function parity
F(x) = f (-x) exists for any x in the domain of function f(x), then f (x) is an even function;
F(x) =-F(x) exists for any x in the domain of function f(x), so f (x) is odd function.
Bian Xiao's recommendation: Summary of the knowledge points of the compulsory examination in high school mathematics.
Properties of (1) odd function and even functions.
I Whether the function is odd function or even function, as long as the function has parity, the domain of the function must be symmetric about the origin.
Ⅱ odd function's image is symmetrical about the origin, and even function's image is symmetrical about the Y axis.
⑵ Thinking about the parity of judgment function
I first determine whether the domain of a function is symmetric about the origin. If it is not symmetric about the origin, then it is a non-odd and non-even function.
Ii. determine the relationship between f(x) and f(-x);
If f(x) -f(-x)=0, or f(x) /f(-x)= 1, the function is even;
If f(x)+f(-x)=0, or f(x)/ f(-x)=- 1, the function is odd function.
3. The maximum value of the function.
⑴ For quadratic function, the collocation method is used to transform the function into the form of y=(x-a)2+b, and the maximum or minimum value of the function is obtained.
⑵ For the function that is easy to draw the function image, draw the image and observe the maximum value from the image.
(3) Maximum value of quadratic function on closed interval
I judge whether the vertex of the quadratic function is within the expected interval. If it is within the interval, then use II to connect it. If it is not within the interval, then use III to connect it.
Two. If the vertex of the quadratic function is within the expected interval, then in the quadratic function y=ax2+bx+c, a >；; 0, the vertex is the minimum value, a; 0 or a
Iii. If the vertex of the quadratic function is not in the sought interval, the monotonicity of the function in the interval is judged.
If the function is increased on [a, b], the minimum value is f(a) and the maximum value is f (b);
If the function decreases in [a, b], the minimum value is f(b) and the maximum value is f(a).
Senior one mathematics 3 basic elementary functions
1, exponential function: function y = ax(a >；; 0 and a ≠ 1) are called exponential functions.
a > 1 0 & lt； a & lt 1
Definition domain x ∈ r
Range y ∈ (0, +∞) y ∈ (0, +∞)
Monotonicity, monotonicity, increasing, monotonicity, decreasing.
Odd non-even function
Through the fixed point (0, 1) (0, 1)
Note: (1) It can be seen from the monotonicity of the function that the maximum value of the exponential function in the closed interval [a, b] is:
A> is in 1, with the minimum value being f(a) and the maximum value being f (b); 0<a< is at 1, with the minimum value being f(b) and the maximum value being f(a).
(2) for any exponential function y = ax(a >；; 0 and a≠ 1), both have f (1) = a.
2. Logarithmic function: function y = logax(a >；; 0 and a ≠ 1) are called logarithmic functions.
a > 1 0 & lt； a & lt 1
The region x ∈ (0, +∞) x ∈ (0, +∞)
Range y ∈ r y ∈ r
Monotonicity, fully defined domain, monotonicity, recursion, monotonicity, decreasing.
Odd non-even function
Pass through the fixed point (1, 0) (1, 0)
3. power function: function y=xa(a∈R). In high school, power function only studies I quadrant.
(1) All power functions are defined in the interval of (0, +∞) and pass through the fixed point (1, 1).
⑵a & gt； At 0, the power function image passes through the origin, and in the interval of (0, +∞), it is increasing function. The greater a, the greater the slope of the image.
⑶a & lt； 0, the power function is a decreasing function in the interval of (0, +∞).
When X is infinitely close to the origin from the right, the image is infinitely close to the positive semi-axis of Y axis;
When y is infinitely close to positive infinity, the image is infinitely close to the positive semi-axis of X axis.
See the next page for the general diagram of power function.
4. Inverse function: exchange x and y of the original function y=f(x) to get its inverse function x=f- 1(y).
The inverse function image and the original function image are symmetrical about the straight line y = X.
The above "Summary of Knowledge Points of Mathematical Functions in Senior One" was compiled and released by Bian Xiao of Senior Three Network. For more information, please pay attention to the high school network.