The proposition of college entrance examination function and equation thought is mainly reflected in three aspects.
(1) Establish functional relations, construct functional models or solve practical problems through equations and equations;
(2) Dealing with the problems of functions, equations and inequalities from the viewpoint of their mutual transformation;
③ Using the ideas of functions and equations to study series, analytic geometry and solid geometry. When we construct the function model, we still pay great attention to the "three quadratic" examination, especially the objective questions, and the big questions are generally slightly more difficult.
Answering skills of mathematical function questions in college entrance examination
logarithmic function
The general form of logarithmic function is that it is actually the inverse of exponential function. Therefore, the stipulation of a in exponential function is also applicable to logarithmic function.
The graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.
The domain of (1) logarithmic function is a set of real numbers greater than 0.
(2) The range of logarithmic function is the set of all real numbers.
(3) The function always passes (1, 0).
(4) When a is greater than 1, it is monotone increasing function and convex; When a is less than 1 and greater than 0, the function is monotonically decreasing and concave.
(5) Obviously, the logarithmic function is unbounded.
exponential function
The general form of exponential function is that from the above discussion of power function, we can know that if X can take the whole set of real numbers as the domain, then it only needs to be done.
You can get:
The domain of (1) exponential function is the set of all real numbers, where a is greater than 0. If a is not greater than 0, there will be no continuous interval in the definition domain of the function, so we will not consider it.
(2) The range of exponential function is a set of real numbers greater than 0.
(3) The function graph is concave.
(4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.
(5) We can see an obvious law, that is, when a tends to infinity from 0 (of course, it can't be equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing.
(6) Functions always infinitely tend to a certain direction on the X axis and never intersect.
(7) The function always passes (0, 1).
Obviously the exponential function is unbounded.
odevity
Generally, for the function f(x)
(1) If any x in the function definition domain has f(-x)=-f(x), then the function f(x) is called odd function.
(2) If any x in the function definition domain has f(-x)=f(x), the function f(x) is called an even function.
(3) If f(-x)=-f(x) and f(-x)=f(x) are true for any x in the function definition domain, then the function f(x) is both a odd function and an even function, which is called an even-even function.
(4) If f(-x)=-f(x) and f(-x)=f(x) cannot be established for any x in the function definition domain, then the function F (x) is neither a odd function nor an even function, which is called an even-even function.
Description: ① Odd and even are global properties of functions, and they are global.
② The domains of odd and even functions must be symmetrical about the origin. If the domain of a function is not symmetric about the origin, then the function must not be an odd (or even) function.
(Analysis: To judge the parity of a function, we should first check whether its domain is symmetrical about the origin, then simplify and sort it out strictly according to the definitions of odd and even, and then compare it with f(x) to draw a conclusion. )
③ The basis for judging or proving whether a function has parity is definition.
Properties of functions and images
The nature of function is the cornerstone of learning elementary function and the key content of college entrance examination.
Reviewing the properties of functions can start with understanding the definitions of monotonicity and parity of functions from two aspects: "number" and "shape", consolidate them in the judgment and proof of the properties of functions, and deepen them in the process of finding the monotonous interval of composite functions, the maximum value of functions and application problems. The specific requirements are:
1. Correctly understanding the definitions of monotonicity and parity of functions can accurately judge the monotonicity of functions and the monotonicity of functions in a certain interval, and skillfully use definitions to prove the monotonicity and parity of functions.
2. Understand the monotonicity and parity of functions from the perspective of the combination of numbers and shapes, deepen the understanding and application of the geometric characteristics of function properties, and summarize the common methods for finding function values and minimum values.
3. Cultivate students to analyze problems from the viewpoint of movement change, and improve their problem-solving ability by mathematical thinking methods such as number-shape replacement, transformation and combination.
This part focuses on the in-depth understanding of the definitions of monotonicity and parity of functions.
Monotonicity of function can only be discussed in the domain of function. The monotonicity of the function y = f (x) in a given interval reflects the changing trend of the function value in the interval, which is the overall property of the function in the interval, but not necessarily the overall property of the function in the definition domain. The monotonicity of the function is aimed at a certain interval, so it is limited by the interval.
To understand the definition of functional parity, we should not only stay on the two equations of f (-x) = f (x) and f (-x) =-f (x), but also make it clear that any x in the domain has f (-x) = f (x) and f (-x) =-f (x). The necessary and sufficient condition for the image of the obtained function f(x) to be symmetric about the straight line x = a is that f (x+a) = f (a-x) holds for any x in the definition domain. The parity of a function reflects the special symmetry of its corresponding image.
The difficulty in this part is the comprehensive application of monotonicity and parity of functions. According to the known conditions, it is very demanding for students to mobilize relevant knowledge and choose appropriate methods to solve problems.