Assuming that f is a valid function, under what conditions will the inverse of f be a function as well?
A function y = f(x) has an inverse function `f^-1(x)` if and only if for each value of x, the function f(x) gives only one value of y. Only a function that is a one-to-one relation has an inverse function.
For example the function `y = sqrt(x)` does not have an inverse if the range is not restricted as the square root of any number can be both a negative number as well as a positive number, i.e. if `y = sqrt x` , for any positive value of x, y can be negative as well as positive.
f(x) has an inverse function `f^-1(x)` if for any value of x the value of f(x) is unique.