Inverse of function of Set

The inverse of a one-to-one (bijective) function f: A → B is the function g: B → A that reverses the mapping of f. It holds the following property −

f(x) = y ⇔ g(y) = x

The function f is called invertible if its inverse function g exists. For a function to be invertible, it must be one-to-one (injective) − meaning no two different inputs map to the same output − and onto (surjective) − meaning every element in the codomain is mapped to by some element in the domain.

The inverse of f is commonly denoted as f⁻¹.

How Inverse Functions Work

When a function f maps element x from set A to element y in set B, its inverse f⁻¹ maps y back to x. The following diagram illustrates this relationship −

Examples

Example 1: An Invertible Function

A function f : Z → Z, defined as f(x) = x + 5, is invertible since it has the inverse function g : Z → Z, defined as g(x) = x − 5.

Verification −

If f(3) = 3 + 5 = 8

Then g(8) = 8 - 5 = 3  ?

g reverses f, so f is invertible.

Example 2: A Non-Invertible Function

A function f : Z → Z, defined as f(x) = x², is not invertible since it is not one-to-one. Both x and −x produce the same output −

f(3)  = 3²  = 9
f(-3) = (-3)² = 9

Two different inputs (3 and -3) give the same output (9).
Since f is not one-to-one, no inverse exists.

Conclusion

A function is invertible only if it is one-to-one (no two inputs share the same output). The inverse function f⁻¹ reverses the original mapping, sending each output back to its corresponding input.