One To One Function

Consider the function f: N -> N, f(x) = x^2.

f = {(1,1), (2,4), (3,9),..}. Here if x1 x2 then f(x1) f(x2). Such a function is called a one to one function.

Consider g: Z -> Z, g(x) = x^2.

Here g = {(0,0), (1,1), (-1,1), (2,4), (-2,4),..}

Taking x1 = 1 and x2 = -1, we have x1 x2 but g(x1) = g(x2). Such a function is not a one to one function.

Definition: A function f: A - > B is said to be a one to one or injective function, if for all x1,x2 A, x1 x2 => f(x1) f(x2).

One to one function examples:

1. f: Z->Z, f(x) = x^3 is a one to one function because for no two values of x, we have the same f(x).

f = {.(-2,-8), (-1,1), (0,0), (1,1), (2,8)..}

2. f: R -> R, f(x) = 2x+3 is also a one to one function. In general all linear functions are always one to one functions.

One to one function inverse:

For a function f: A-> B, if there exists a function g: B - > A such that gof = and fog = then g: A -> B is called an inverse function of f: A -> B.

For any function f: A-> B, its inverse function, if it exists is unique. Inverse of a function is denoted by the symbol :

Theorem: The necessary and sufficient condition for the existence of the inverse of f: A -> B is that f is one to one and onto i.e. if the inverse of f: A -> B exists, then f is one to one and onto and vice versa.

One to one function graph:

We can know from the graph of the function if the function is one to one or not. If we draw a horizontal line on the graph of any function and if the line cuts the function in only one point, then its one to one otherwise its many to one function. This is called the horizontal line test.

by: nayaknandan

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