Suppose a set of ordered pairs defines a relation. Then the inverse relation is obtained by reversing the coordinates of each ordered pair. This means if (a, b) is an element of a relation then (b, a) is the corresponding element in the inverse relation. In short inverse relation switches input and output. If both the relation and its inverse are functions then two functions serve as inverse functions of each other.

The function g which is the inverse of f can be denoted as f-1 and read as "f inverse".

The Domain and Range of function switch the roles as Range and Domain of the inverse function. This means all functions do not have inverses. For an inverse function to be defined the original function needs to be one-to-one, that is no image in the range can have more than one pre image in the domain.
Definition: Suppose f(x) is an one-to-one function with domain D and range R. Then its inverse function f-1(x) has domain R and range D and defined by
f-1(y) = x ⇔ f(x) = y for any y ∈ R.

  1. Suppose f and f-1 are inverse functions. Then f(a) = b if and only if f-1(b) = a.
  2. The domain and range of a function switch the roles as range and domain for the inverse functions.
  3. Two functions f and g are inverse functions of each other if and only if both the compositions are identity functions.
    [f o g](x) = x and [g o f](x) = x.
  4. The graph of f-1 is the reflection of the graph of f over the line y = x.
  5. If a function is one-to-one, then any horizontal line drawn will intersect the graph of the function at not more than one point. This is property is used in horizontal line test to determine whether the inverse of a function exists.