幂函数习题精选精讲 (3)

幂函数习题精选精讲1（Selected exercises of power function 1）power function

As the main function of high school mathematics, throughout the high school mathematics learning always, and the power function is a part of which, though it simple, but contains some important mathematical ideas. The analysis of several cases, in order to expand students' thinking.

First, the idea of classified discussion

Example 1, the image of the known function and the two axis have no common point, and its image about the Y axis symmetry, the value of N, and draw the function of the image

Solution: since there is no common point between the image and the Y axis, the image is symmetrical about the Y axis, even if it is even

At that time, not even;

At that time, even numbered;

At that time, even numbered;

At that time, not even;

At that time, even numbered;

So n is, 1 or 3.

At this point, the power function is interpreted as or, and its image is shown in Figure 1

Two, the idea of combining numbers and shapes

Example 2 the known point is on the image of the power function, points on the image of the power function

When asked why x is worth: (1); (2); (3)

Analysis: by the definition of power function, first find and parse the formula, and then use the image to judge

Solution: set by you to,

So, that is. Then by order, you have to,

So, that is made. In the same coordinate system

And the image, as shown in Figure 2, is illustrated by the image:

(1) when or when,;

(2) at that time,;

(3) when and when

Summary: the combination of numbers and shapes has important applications in the discussion of inequality, and pays attention to the implicit conditions in this topic

Three, the mathematical idea of transformation

Example 3 the domain of the function is all real numbers, and the range of the real number m is ()

A.

B.

C.

D.

Resolution: if the domain of the function is all real numbers, it can be converted to all real numbers, that is, and

B

Three kinds of discussion questions in power function

The so-called classification discussion, the essence is "to crush one by one, break up the whole into parts, zero for the whole plot strategy. Classification discussion should focus on understanding and mastering the principles, methods and skills to determine the object classification, all clear classification standards, classification discussion is not heavy, no leakage. In power function, classification discussion the idea of an important embodiment, according to the image and the nature of power function, on the basis of monotonity power function are discussed, the results can be achieved.

Type 1: parameter range

Example 1, the known function is even function, and the value of the m, and determine the analytic expression

Analysis: the function is the even function, has already limited to even number, and, as long as according to the condition classification discussion, then can obtain the value of M, and thus determine the analytic formula

Solution: dreams is an even function, R is even.

And dreams, namely, finishing, dry, dry.

And dreams, or 1. star

When m=0 is odd (down); at that time, even.

Therefore, the value of M is 1

Commentary: when using the classified discussion idea to solve the problem, we should fully excavate each information in the known condition, and do not weigh or leak, so that we can lay a solid foundation for solving the problem correctly

Type two: solving the problem of existence

Example 2, known functions, set functions, ask whether the existence of real numbers, so that in the interval is less than the function, and in the interval is an increase function If

it exists, ask for it; if not, please explain the reason

Analysis: when judging the monotonicity of the function, we can use the definition, but also combine the image and the nature of the function to judge, but we should pay attention to the determination of the symbol in the problem, it depends on the value range of the independent variable

Solution: dreams, then.

Assume the existence of real numbers so as to satisfy the problem setting condition,

Suppose that

.

If, in the set, is a decreasing function, it should be the permanent establishment.

Dreams, and perylene,

Star.

Thus the constant will be established,

Yes, that is

If, in is to make easy, increasing function is constant should be established.

Dreams,,

So, while, dry.

If you want to be established, then there will be

To sum up, we can see that there is a real number, so that on the upper is the decreasing function, and on the other is the increasing function

Comment: This is a comprehensive subject, is a comprehensive application of the nature of power function. The monotonicity of the judgment function, from the definition, can also be judged according to the function and nature of the image, but the higher requirement on the ability to analyze and solve problems, which usually pay attention to targeted training.

Type three: the nature of an analogous power function to discuss the change in function values

Example 3 discusses the change in function values when the function increases with the X

Analysis: we should first determine whether the function is a constant function, and then look at the power exponent, and discuss the nature of the power function

Solution: (1) when, if, as the constant function;

(2) at that time, or, at this point, the function is a constant function;

(3) when the function is the decreasing function, the function value decreases with the increase of x;

(4) the function is the increasing function immediately or in time, and the function value increases with the increase of x;

(5) when the function is an increasing function, the function value increases with the increase of x;

(6) when the function is a decreasing function immediately, the function value decreases with the increase of X

Commentary: the problem of parameter coefficients can be said to be a fatal killer in solving problems and an important factor leading to errors. This should arouse our high vigilance

Power function exercises

The power function of the knowledge points, on the surface and less content easy, actually otherwise. It contains the combination, classification discussion, transformation of mathematical thought, is a good carrier to cultivate the students' mathematical thinking ability. Following the application method of a changeful exploration of the nature of power function.

Example 1 if you try to find the real value range of M

Delusiveness (Shuxingjiege): we can see from Figure 1

The solution is correct

Analysis: Although the functions are monotone in interval and on the other, but they are not monotonic in the interval, so it is wrong to use monotone solution

Positive solutions (classification, discussion):

(1)

Xie jie;

(2) there is no solution at this time;

(3) the solution is obtained

In summary, available

Now, change the index in example 1 to 3 and see what happens

Example 2 if you try to find the real value range of M

We (discuss): known by the figure 2,

(1) 1;

(2) there is no solution at this time;

(3) the solution is obtained

In summary, available

Analysis: it is obvious that this method mechanically imitates the correct solution of example 1, but ignores the difference of domain between functions, which makes us feel the important role of the domain of power function in solving problems

Positive solution (using monotonicity): because the function is monotonically increasing, so the solution is obtained

Example 2. The correct solution deepens the understanding of the monotonicity of power functions and activates the thinking of students. Here are two more questions and Solutions

Example 3 if you try to find the real value range of M

Solution: as shown in Figure 3, the solution is obtained

Example 4 if you try to find the real value range of M

Analysis: make the power function of the image in figure 4. by the image in this function is not necessarily monotonic in, if the classification step is discussed more complex, the problem is transformed into a monotone interval is the key. When considered. Then there are, namely.

And the power function is monotone increasing in dreams,

So, solution, or M > 4.

The above solution realizes the increasing of the power function in the first quadrant, and then skillfully applies the

transformation thinking to solve the problem, thus avoiding the classified discussion and deepening the thinking of the students

Solving problems: through the above research, we have a deeper understanding of the domain, monotonicity, parity and image of power function,

At the same time, the solution to the inequality of shape (such as constant) has the following experience:

(1) when the solution is the same as example 1

(2) when the solution is the same as example 2

(3) when the solution is the same as example 3

(4) when the solution is the same as example 4.

Editor's comment: This paper through a variety of transformation for a typical example, enable us to power function properties and image have a more profound understanding of the 4 cases, although in the process of solving problem involves the solution inequality with absolute value, beyond the scope of our study, but it contains the "transformation" a thought, widen our problem-solving ideas, but also reflects the flexibility of the application ability of knowledge, of course, this problem can also discuss the method of solution, the students can have a try.