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关于函数值域的确定,是统编高中数学教材中的一个难点。学生作题通常没有一般方法可循,并且容易出现混乱和错误。本文拟给出求初等函数值域的一般方法。下面我们提出三个定理,为尽量避免使用较多的实数理论,仅用几何图形加以直观说明,不给出严格论证。然后归纳出只需运用简单的导数知识,对中学生可行的初等函数值域的一般方法。定理1 若函数y=f(x)满足条件 (1)、在闭区间〔a,b〕上连续; (2),最大值、最小值分别为M,m,则函数y=f(x)的值域为〔m,M〕。(其中mM) 定理1中,M、m的存在性与结论的正确性从函数图象(图1)上看是很明显的。例1,求函数f(x)=x~2-5x+6,x∈〔2,
The determination of the function value range is a difficult point in the compilation of high school mathematics textbooks. There is usually no general way for students to write questions, and it is prone to confusion and mistakes. This article proposes a general method to find the range of elementary functions. In the following, we propose three theorems. In order to avoid using more real number theory, we only use geometric figures to explain it directly and do not give strict arguments. Then we generalize the general method of elementary function range that can be applied to middle school students simply by using simple derivative knowledge. Theorem 1 If the function y=f(x) satisfies condition (1) and is continuous on the closed interval [a,b]; (2), the maximum and minimum values are M,m respectively, then the function y=f(x) The range is [m,M]. In (The mM) Theorem 1, the existence of M and m and the correctness of the conclusion are obvious from the function image (Fig. 1). Example 1, find the function f(x)=x~2-5x+6, x∈[2,