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求函数的值域涉及到的知识面很广,是教学中的难点之一,笔者在教学中教给学生用下列方法求函数的值域,取得了理想的效果。 一、运用方程的思想求函数值域 运用方程的思想求函数值域,就是将函数y=f(x)的解析式视为关于x的方程(y为参数),只需根据方程有实数解的条件,求出使该方程在函数定义域内有解的所有y值的集合,则此集合目即为函数y=f(x)的值域。 例1 求函数y=5x-1/2x-3(x∑R,且x≠3/2)的值域, 解:把函数式看成关于x的方程,变形得 (2y-5)x=3y+1, 由此可见,原方程在函数定义域内有解的充要条件是2y-5≠0,即y≠5/2,从而可确定所求函数的值域为(-∞,5/2)U(5/2,+∞)。
Finding the value range of a function involves a wide range of knowledge, which is one of the difficulties in teaching. In the teaching, the author teaches students to use the following methods to find the value range of the function, and achieves an ideal result. First, the use of the idea of the equation to find the value of the function domain The use of the idea of the equation to find the function range, that is, the analytical function of the function y = f (x) as an equation about x (y is a parameter), only need to have a real solution according to the equation The condition is to find the set of all y-values that make the equation have a solution in the domain of the function, then this collection is the domain of the function y=f(x). Example 1 Find the range of the function y=5x-1/2x-3(x∑R, and x≠3/2). Solution: Consider the functional formula as an equation about x, and the result is (2y-5)x= 3y+1, It can be seen that the necessary and sufficient condition for the original equation to have a solution in the function definition domain is 2y-5≠0, ie y≠5/2, so that the range of the function to be evaluated can be determined as (-∞,5/ 2) U(5/2, +∞).