Value range and solution of difficult function
The value range of function and its solution are one of the key contents of college entrance examination in recent years. This section mainly helps the examinee to flexibly master various methods of value range, and be able to use the value range of functions to solve practical application problems.
● Difficult magnetic field
(★★★★★) Let m be a real number, record M={m | m>1}, f (x)=log3 (x2-4mx+4m2+m+).
(1) It is proved that f (x) is meaningful for all real numbers when m ∈ M; Conversely, if f (x) has meaning for all real numbers x, then m ∈ M.
(2) When m ∈ M, find the minimum value of function f (x).
(3) Proof: For each m ∈ M, the minimum value of function f (x) is not less than 1.
Difficulty Parity and Monotonicity (I)
The monotonicity and parity of functions are one of the key contents of the college entrance examination, and the examination contents are flexible and diverse. This section mainly helps the examinee to understand the definition of parity and monotonicity, master the determination method, and correctly understand the image of monotone function and odd even function.
● Difficult magnetic field
(★★★★) Let a>0, f (x)=an even function on R, and (1) find the value of a; (2) It is proved that f (x) is an increasing function on (0,+∞).
Difficulty Parity and Monotonicity (II)
The monotonicity and parity of functions are one of the key and hot topics in the college entrance examination, especially the application of gender. This section mainly helps the examinee learn how to use the gender quality to solve problems, master the basic methods, and form a sense of application.
● Difficult magnetic field
(★★★★★) It is known that the even function f (x) is an increasing function on (0,+∞), and f (2)=0, the solution inequality f [log2 (x2+5x+4)] ≥ 0.
● Case study
[Example 1] It is known that the odd function f (x) is a subtraction function defined on (- 3,3) and satisfies the inequality f (x-3)+f (x2-3)<0. Let the solution set of the inequality be A, B=A →{x | 1 ≤ x ≤}, and find the maximum value of the function g (x)=- 3x2+3x-4 (x ∈ B).
Difficulties: exponential function and logarithmic function
Exponential function and logarithmic function are one of the key contents of the college entrance examination. This section mainly helps candidates master the concepts, images and properties of the two functions and use them to solve some simple practical problems.
● Difficult magnetic field
(★★★★) Set f (x)=log2, F (x)=+f (x).
(1) Try to judge the monotonicity of function f (x), and use the monotonicity definition of function to give proof;
(2) If the inverse function of f (x) is f-1 (x), it is proved that for any natural number n (n ≥ 3), there is f-1 (n)>;
(3) If the inverse function F-1 (x) of F (x), it is proved that the equation F-1 (x)=0 has a unique solution.
