One-one function

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A function f is called one-one function if it never adopts the same value twice. This means, one-one functions return a unique range for each element of their domain.

Other names given to the one-one function are one to one, or injective function.

One-one functions are a set of functions which denote the relation between sets, elements or identities. The other two sets are surjective functions and bijective functions.

Definition

The function f(x) is called one-one function when for every value of x in the domain of the function, there will be a unique value of f(x).

Geometric proof

Similar to the vertical line test (VLT) for functions, there is a horizontal line test (HLT) to prove the one-one property. A function is one-one if and only if no horizontal line intersects its graph more than once.


In the graph below, the function y=x^3 is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.

One-one function geometric proof true.PNG


In the graph below, the function y=x^2 is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.

One-one function geometric proof.PNG

Properties

  • The domain of f equals the range of f-1.
  • f^{-1}(f_(x))=x for every x in the domain of f and f
  • The graph of a function and the graph of its inverse are symmetric with respect to the line y=x.
  • If f and g are both one-one, then f°g follows injectivity.
  • If g°f is one-one, then function f is one-one, but function g may not be.
  • A one-one function is either strictly decreasing or strictly increasing.
  • A function that is not a one-one is considered as many-to-one.
  • Parabolic functions are not one-one functions.

Examples

Examples of one-one functions include:

  • Identity function: f(x) is always injective.