What is an inverse function

The inverse function is a function in mathematics. Suppose the domain of function y=f(x) is D and the value domain is f(D); if for every y in the value range f(D), there is and is only one x in D such that g(y)= x, then according to this corresponding rule, a function defined on f(D) is obtained, and this function is called the inverse function of the function y=f(x).

What is an inverse function?

Generally speaking, assuming that the value range of function y=f(x)(x∈A) is C, if a function g(y) is found, g(y) will be Equal to x, such a function x= g(y)(y∈C) is called the inverse function of the function y=f(x)(x∈A), recorded as y=f^(-1)(x). The domain and domain of the inverse function y=f ^(-1)(x) are respectively the domain and domain of the function y=f(x). The most representative inverse functions are logarithmic functions and exponential functions.

Generally, if x and y correspond to a certain correspondence relationship f(x), y=f(x), then the inverse function of y=f(x) is x=f(y) or y=f﹣¹(x). The condition for the existence of an inverse function (default is a single-valued function) is that the original function must have a one-to-one correspondence (not necessarily in the entire number field). Note: The superscript "−1" does not refer to power.

Extended information: Properties of inverse functions

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

(2) The necessary and sufficient condition for the existence of an inverse function of a function is that the domain and value domain of the function are one-to-one mapping;

(3) A function and its inverse function The monotonicity is consistent in the corresponding interval;

(4) Most even functions do not have inverse functions (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 an even function and has an inverse function. The domain of its inverse function is {C} and the value range is {0}). An odd function does not necessarily have an inverse function. When intercepted by a straight line perpendicular to the y-axis, it can pass through 2 or more points, that is, there is no inverse function. If an odd function has an inverse function, then its inverse function is also an odd function.

(5) The monotonicity of a continuous function is consistent within the corresponding interval;

(6) A function that strictly increases (decreases) must have an inverse function that strictly increases (decreases) ;

(7) Inverse functions are mutual and unique;

(8) The definition domain and value domain are opposite, and the corresponding rules are mutually inverse (three inversions);

(9) Derivative relationship of inverse function: If x=f(y) is strictly monotonic and differentiable on the open interval I, and f'(y)≠0, then its inverse function y=f -1(x) is in It can also be differentiated within the interval S={x|x=f(y),y∈I};

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

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