How to do the inverse function problem: determine the range of the original function. From the expression of the original function, find the "expression of X about Y". Swap x and y and attach fields.

Let the function y=f(x) have a domain of d and a range of f(D). If there is only one X in D for each Y in the interval f(D) so that g(y)=x, a function defined on f(D) is obtained according to this corresponding rule, which is called the inverse function of function y=f(x).

Generally speaking, if X and Y correspond to a corresponding relation f(x) and y=f(x), the inverse function of y=f(x) is x=f- 1(y). The inverse function (single-valued function by default) exists on the condition that the original function must correspond to each other.

Compared with the inverse function y=f- 1(x), the original function y=f(x) is called a direct function. The images of the inverse function and the positive function are symmetrical about the straight line y = X, because if (a, b) is any point on the image of y=f(x), that is, b=f(a).

According to the definition of inverse function, there is a=f- 1(b), that is, point (b, a) is on the image of inverse function y=f- 1(x). While points (a, b) and (b, a) are symmetrical about the straight line y = X. From the arbitrariness of (a, b), we can know that f and f- 1 are symmetrical about y = X.