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求函数值域的方法很多,如观察法、配方法、判别式法、反函数法、换元法等等。这些方法如使用得当,可使问题迅速获得解决;如果使用不当,往往会导致谬误。现将几种常见错误举例辨析于下: 一、使用配方法时,因所配得的完全平方式等于零时无实数解而导致错误。例1 已知x>0,求函数 y=x+1/x+9的值域。错解: y=x+1/x+9=(x~(1/2)+1/(x~(1/2)))~2+7。∵ (x~(1/2)+1/(x~(1/2))~2≥0,∴y≥7。
There are many ways to find the range of functions, such as observation method, matching method, discriminant method, inverse function method, substitution method, and so on. If these methods are used properly, problems can be solved quickly; if they are used improperly, they will often lead to delays. Several common mistakes are identified in the following examples: First, when using the matching method, there is no real solution because the fully-flattened method is equal to zero, resulting in an error. Example 1 Known x>0, find the range of the function y=x+1/x+9. The wrong solution: y=x+1/x+9=(x~(1/2)+1/(x~(1/2)))~2+7. ∵ (x~(1/2)+1/(x~(1/2))~2≥0, ∴y≥7.