巧解抽象函数的奇偶性、对称性和周期性问题

Skillfully Solving the Parity, Symmetry, and Periodicity Problems of Abstract Functions

抽象函数没有具体的解析式,只给出函数满足一部分的运算法则或者性质。抽象函数综合体现学生数学抽象、逻辑推理、直观想象和数学运算等数学核心素养。本文从函数的对称性角度推导函数周期性,在抽象函数给出满足对称性的条件下,通过函数的轴对称和中心对称点“知二求一”。利用周期性轻松解决近十年高考中抽象函数求值和求和问题,巧用抽象函数的对称性满足的等式反推选项对称性是否正确。使得晦涩难懂函数问题解题思路清晰明了和简单易懂。

Abstract functions do not have specific analytical expressions, but only provide some operational rules or properties that the function satisfies. Abstract functions comprehensively reflect students’ core mathematical literacy such as mathematical abstraction, logical reasoning, intuitive imagination, and mathematical operations. This article deduces the periodicity of a function from the perspective of its symmetry. Under the condition that the abstract function satisfies symmetry, the function is known as “knowing two and finding one” through its axial symmetry and central sym-metry points. It is easy to solve the problem of abstract function evaluation and summation in the college entrance examination in the past ten years by using periodicity, and deduce whether the symmetry of the option is correct or not by using the equation that the symmetry of the abstract function satisfies. It makes the solution of obscure function problems clear and simple to understand.