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人们用方程组求方程f(x)=f~(-1)(x)的解,f(x)与f~(-1)(x)的图象的交点,为什么有的无解、有的漏解呢?由于前者结果是后者交点的横坐标,所以这两类题的解都由f(x)与f~(-1)(x)的图象是否相交、交点在何处来确定。要回答上述疑问和有依有据地解好这两类问题,必须找到,f(x)与f~1(x)的图象相交的规律。笔者探索这规律时,发现连续函数有下面几个有趣的命题(命题中的函数连续)。[命题1] 斜率不为±1的一次函数与其反函数的图象有且仅有一个交点,这个交点在直线y=x上。
People use the system of equations to find the solution of the equation f(x)=f~(-1)(x), the intersection of the image of f(x) and f~(-1)(x), and why some solutions are not available. What is the leakage? Because the former result is the abscissa of the latter’s intersection point, the solutions of the two types of questions are determined by the intersection of the images of f(x) and f~(-1)(x) and the point of intersection. determine. To answer these questions and solve these two types of questions in a well-founded manner, we must find the law of intersection of the images of f(x) and f~1(x). When the author explores this law, he finds that the continuous function has the following interesting propositions (functions in the proposition are continuous). [Proposition 1] The image of the first-order function whose slope is not ±1 has only one intersection with the image of its inverse function. This intersection is on the straight line y=x.