One-one function: Difference between revisions

Latest revision as of 04:12, 28 April 2022

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.

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.

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.

@@ Line 1: / Line 1: @@
-A [[function]] <math>f</math> is called one-one function if it never adopts the same value twice.
+A [[function]] <math>f</math> 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 function]]s and [[bijective function]]s.
+== Definition ==
+The function <math>f(x)</math> is called one-one function when for every value of <math>x</math> in the domain of the function, there will be a unique value of <math>f(x)</math>.
 == Geometric proof ==
-A function is one-one if and only if no horizontal line intersects its graph more than once.
+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 <math>y=x^3</math> is intersected once by the horizontal line. Therefore the function is geometrically proven to be one-one.
+[[File:One-one function geometric proof true.PNG|350px]]
+In the graph below, the function <math>y=x^2</math> is intersected twice by the horizontal line. Therefore the function is geometrically proven no to be one-one.
+[[File:One-one function geometric proof.PNG|350px]]
+== Properties ==
+* The domain of f equals the range of f<sup>-1</sup>.
+* <math>f^{-1}(f_(x))=x</math> 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 function]]s are not one-one functions.
+== Examples ==
+Examples of one-one functions include:
+* Identity function: f(x) is always injective.