Definition: a function with the shape of y = x a (a is constant), that is, a function with the base as the independent variable and the exponent as the dependent variable is called a power function.

Domain and Value Domain:

When a is a different numerical value, the different situations of the domain of the power function are as follows: if a is any real number, the domain of the function is all real numbers greater than 0; If a is a negative number, then X must not be 0, but the definition 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 definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0. When x is different, the range of power function is different as follows: when x is greater than 0, the range of function is always a real number greater than 0. When x is less than 0, only when 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.

Nature:

For the value of a nonzero rational number, it is necessary to discuss their respective characteristics in several cases:

First of all, we know that if a=p/q, q and p are integers, then x (p/q) = the root of q (p power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞). When the exponent n is a negative integer, let a=-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.

Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;

Rule out the possibility of 0, that is, for X.

The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.

2. Senior one mathematics compulsory three knowledge points induction

(1) Algorithm concept: Mathematically, an "algorithm" in the modern sense usually refers to a kind of problem that can be solved by a computer as a program or step, and these programs or steps must be clear and effective and can be completed in a limited number of steps.

(2) Features of the algorithm:

① finiteness: the sequence of steps of the algorithm is finite and must stop after a finite operation, not infinite.

(2) Certainty: Every step in the algorithm should be certain, which can be effectively executed and get certain results, and should not be ambiguous.

③ Sequence and correctness: The algorithm starts from the initial step and is divided into several definite steps. Each step can only have one definite subsequent step, and the former step is the premise of the latter step. Only when the previous step is finished, can the next step be carried out, and every step is done correctly, and the problem can be completed.

④ inevitability: the solution of a problem is not necessarily true, and there are different algorithms for a problem.

⑤ Universality: Many specific problems can be solved by designing reasonable algorithms, such as mental calculation and calculator calculation, which must be solved by limited and pre-designed steps.

3. The induction of three compulsory knowledge points in senior one mathematics.

1. Division is a method to find the common divisor. This algorithm was first proposed by Euclid around 500 BC, so it is also called Euclid algorithm.

2. The so-called phase shift method is to divide the larger number by the smaller number for a given two numbers. If the remainder is not zero, the smaller number and the remainder form a new pair of numbers, and continue the above division until the larger number is divided by the decimal, then the divisor is the common divisor of the original two numbers.

3. Multiphase subtraction is a method to find the common divisor of two numbers. The basic process is: for two given numbers, subtract the smaller number from the larger number, then compare the difference with the smaller number and subtract the number from the larger number. Continue this operation until the numbers obtained are equal, and this number is the common divisor.

4. Qin algorithm is a method to calculate the value of univariate quadratic polynomial.

5. The commonly used sorting methods are direct insertion sorting and bubble sorting.

6. The carry system is an agreed counting system for the convenience of counting and operation. "All in one" is a K-base system, and the base of the base system is K.

7. The method of converting decimal number into decimal number is: first, write decimal number as the sum of the product of the number on each bit and the power of k, and then calculate the result according to the operation rules of decimal number.

8. The method of converting decimal numbers into decimal numbers is: divide by k and take the remainder. That is, k is used to continuously remove the decimal or the quotient until the quotient is zero, and then the remainder obtained each time is arranged backwards into a number, which is the corresponding decimal.

4. The induction of three compulsory knowledge points in senior one mathematics.

1. Some basic concepts: (1) vector: a quantity with both magnitude and direction.

(2) Quantity: only size, no directional quantity.

(3) Three elements of a directed line segment: starting point, direction and length.

(4) Zero vector: a vector with a length of 0.

(5) Unit vector: a vector with a length equal to 1 unit.

(6) Parallel vector (* * * line vector): non-zero vector with the same or opposite direction, and zero vector is parallel to any vector.

(7) Equal vectors: vectors with equal length and the same direction.

2. Vector addition operation:

⑵ The characteristics of triangle rule: end to end.

⑵ Characteristics of parallelogram rule: * * starting point

5. The induction of three compulsory knowledge points in senior one mathematics.

First, the basic steps of finding the moving point trajectory equation. 1. Establish an appropriate coordinate system and set the coordinates of the dispatching point m;

2. Write a set of points m;

3. List the equation = 0;

4. Simplify the equation to the simplest form;

5. check.

Second, the common methods to solve the trajectory equation of the moving point: There are many methods to solve the trajectory equation, such as literal translation, definition, correlation point method, parameter method, intersection method and so on.

1. Literal translation method: the conditions are directly translated into equations, and the trajectory equation of the moving point is obtained after simplification. This method of solving trajectory equation is usually called literal translation.

2. Definition method: If it can be determined that the trajectory of the moving point meets the definition of the known curve, the equation can be written by using the definition of the curve. This method of solving trajectory equation is called definition method.

3. Correlation point method: use the coordinates x and y of moving point Q to represent the coordinates x0 and y0 of related point P, then substitute them into the curve equation satisfied by the coordinates (x0, y0) of point P, and simply sort them out to get the trajectory equation of moving point Q.. This method of solving trajectory equation is called correlation point method.

4. Parametric method: When it is difficult to find the direct relationship between the coordinates x and y of the moving point, the relationship between x and y and a variable t is often found first, and then the equation is obtained by eliminating the parameter variable t, that is, the trajectory equation of the moving point. This method of solving trajectory equation is called parameter method.

5. Trajectory method: eliminate the parameters in the two dynamic curve equations, and get the equation without parameters, which is the trajectory equation of the intersection of the two dynamic curves. This method of solving trajectory equation is called trajectory method.

The general steps of finding the moving point trajectory equation;

(1) Establish a system-establish a suitable coordinate system;

② set point-set any point on the trajectory P(x, y);

(3) Formula —— List the relationship that the moving point P satisfies;

④ Substitution-according to the characteristics of conditions, the distance formula and slope formula are selected, converted into equations about X and Y, and simplified;

⑤ Proof —— Prove that the equation is a moving point trajectory equation that meets the requirements.