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解方程的过程中,由于利用了方程变形,可能使未知数的允许值的集合的扩大或缩小,因此可能导致解的增加或遗失,同学对于这些问题的认识是较模糊的。在解方程后往往用代入法进行检验,有时会增加计算上的麻烦。必须强调:在解方程的过程中应了解产生解的增加或遗失的原因,检验时,除了代入原方程进行验算外,还需利用方程的定义域的知识来检验,教师在课堂教学中应指导同学如何将同解理论贯彻到解方程中去。在课本中虽然是叙述了一些关于方程的同解问题,在解指数方程和对数方程中仅依靠这些知识还是感到不足,为此提出如下的见解。 (一)为了使同学更深刻地去理解同解方程的理论,要求同学明确如下几个概念是很有必要的。 (1)未知数的允许值的集合。使函数f(x)有意义的x的值,叫做函数f(x)的未知数的允许值,这些允许值的全体,叫做函数f(x)
In the process of solving the equation, due to the use of the equation deformation, the set of allowable values of the unknown may be expanded or reduced, which may lead to increase or loss of the solution. The students’ understanding of these problems is rather vague. After the equations are solved, they are often tested by substituting them, which sometimes increases the computational trouble. It must be emphasized that in the process of solving the equation, the cause of the increase or loss of the solution should be understood. When testing, besides the substitution of the original equation for verification, the knowledge of the domain of equations should be used to test, and the teacher should guide the classroom teaching. How will the students implement the same solution theory into the equation? Although in the textbook, some problems concerning the solution of the equations are described, and it is still insufficient to rely only on these knowledge in solving exponential equations and logarithmic equations, the following insights are proposed. (1) In order for students to understand the theory of solution equations more profoundly, it is necessary for the students to clarify the following concepts. (1) The set of allowed values of unknowns. The value of x that makes the function f(x) meaningful is called the allowable value of the unknown of the function f(x). The total of these allowable values is called the function f(x).