How to prove that a function has an inverse function and find out its inverse function? Take y = (x ^ 2-1) / (x ^ 2 + 1) as an example

How to prove that a function has an inverse function and find out its inverse function? Take y = (x ^ 2-1) / (x ^ 2 + 1) as an example

If and only if there is a one-to-one mapping between the domain of definition and the domain of value, a function has an inverse function
In y = (x ^ 2-1) / (x ^ 2 + 1), the definition field is R. when x = 1 or - 1, there is y = 0, so y = (x ^ 2-1) / (x ^ 2 + 1) has no inverse function
If x > 0 (or any mapping between the domain and the range) is restricted, then y = (x ^ 2-1) / (x ^ 2 + 1) has an inverse function
To find the inverse function is to do two things: to solve X and express it with y; 2. To find the range of value: to find the range of the original function, that is, the domain of definition of the inverse function