Inverse function is a function in mathematics. Let the definition domain of function y=f(x) be d and the value domain be 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).

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Properties of inverse function

The image of (1) function f(x) and its inverse function f-1(x) is symmetric about the straight line y=x;

(2) The necessary and sufficient condition for a function to have an inverse function is that the domain of the function is mapped to the domain of the function one by one;

(3) A function and its inverse function are monotonic in the corresponding interval;

(4) Most even functions have no inverse function (when the function y=f(x), the domain is {0}, and f(x)=C (where c is a constant), then the function f(x) is even and has an inverse function, and the domain of the inverse function is {C} and the range is {0}). Odd function doesn't necessarily have an inverse function. When it is cut by a straight line perpendicular to the Y axis, it can pass through two or more points, that is, there is no inverse function. If a odd function has an inverse function, its inverse function is also odd function.

(5) The monotonicity of continuous functions is consistent in the corresponding interval;

(6) The strict increase (decrease) function must have the inverse function of strict increase (decrease);

(7) Inverse functions are mutually unique;

(8) Definition domain and value domain are opposites, and the corresponding laws are reciprocal (three opposites);

(9) Derivative relation of inverse function: If x=f(y) is strictly monotonic and differentiable in the open interval I, and f'(y)≠0, then its inverse function y=f-1(x) is in the interval S = {x | x = f (y), y ∈.

(10) The inverse function of y = x is itself.