The word "function" comes from a root that means "to do" or "to perform." Accordingly, a function is something that does something to a number or performs some operation on it. The function f(x) = x + 8 adds 8 to some unknown number x. If x happens to be 13, this function adds 8 to it, so that the sum is 21.
An inverse function is still a function. It still performs some operation on a number. However, from the perspective of the original function to which it is an inverse, it does things backwards. For example, the function g(x) = x - 8 is the inverse function of f(x) = x + 8. Instead of adding 8 to a number as f(x) did, g(x) subtracts 8. Accordingly, if g(x) operates on the number 21, it will undo the work of f(x) and reduce the numerical value to 13, as it was before f(x) performed its operation.
All inverse functions are one-to-one functions. This means that when the inverse function g(x) = x -- 8 performs its characteristic operation, each value of x yields only one answer. If the function operates on the number 21, the answer will always be 13. However, a function operating on the square root of x is not an inverse function because it can yield a positive or a negative value. For example, the square root of 25 can be either 5 or -5. However, in the set of all positive real numbers, a function that operates on the square root of x is an inverse function to the function operating on x squared, because negative numbers do not occur in this set.
In some cases, the graph of an inverse function shows a continuous increase in value. As x increases, so does y. In the case of other inverse functions, the y values continuously decline as x values increase. If the y values first increase and then decrease, the function is not an inverse function. A horizontal line intersects the graph of an inverse function at only one point.