Injective Function

An injective function is a function that connects an element of one set to a distinct element of another set. An injective function is another name for a one-to-one function. Injective functions can be found in a variety of contexts. The name and roll number of a student in a class, as well as the person and his shadow, are all examples of injective function.

An injective function f(.) has the property that any two distinct values x1,x2 in the domain have different function values  f(x1)≠f(x2). So, if you have some value f(x) in the co-domain, you know that the domain has precisely one value x as pre-image. As a result, there is the alternative concept of one-to-one for an injective function.

The injective function can be expressed as an equation or as a set of items. It is a one-to-one function, f(x) = x + 5. This can be understood by considering the function’s domain items to be the first five natural integers. The injective function f = (1, 6, 2), (2, 7), (3, 8), (4, 9), (5, 10)

What is injective function

A function is one-to-one or injective if it does not map two different elements in the domain to the same element within the range

Every element of a given set is associated to a separate element of another set in an injective function. A one-one (or injective) function  f : X → Y is defined as such if the images of separate elements of X under f are unique, i.e., for every x1, x2 ∈ X, there exists distinct y1, y2 Y, such that , f(x1) = y1, and fx2=y2

The Venn diagram illustrations above assist in quickly locating and comprehending the injective function. We can see that each element of set X is mapped to a distinct element of set Y. Furthermore, if any element in set Y is an image of more than one element in set X, the function is not one-to-one or injective.

Injective Function Properties

The following are a few key characteristics of injective functions.

• An injective function’s domain and range are equivalent sets.

• The sets representing the injective function’s domain and range have the same cardinal number.

• Injective functions are always represented as a straight line on a graph.

• The injective function has the properties of being reflexive, symmetric, and transitive.

Examples of injective function

Example: The two functions f(x) = x + 1 and g(x) = 2x + 3 are one-to-one. Determine gof(x), as well as whether or not this function is an injective function.

Solution: 

The functions offered are f(x) = x + 1 and g(x) = 4x + 5. To find gof, we must mix these two functions (x).

g(f(x)) = g(x + 1) = 4(x + 1) + 5 = 4x + 4+ 5= 4x + 9

gof(x) = 4x + 5

Now consider the domain of this composite function to be the first five natural numbers.

gof1= 41+ 9 = 4 + 9 =13

gof(2) = 4(2) + 9 = 8 + 9 = 17

gof(3) = 4(3) + 9= 12+ 9= 18

gof(4) = 4(4) + 9 = 16 + 9 = 25

gof(5) = 4(5) + 9= 20 + 9 = 29

gof(x) = {(1, 13), (2, 17), (3, 18), (4, 25), (5, 29)

In this case, the distinct element in the function’s domain has a distinct image in the range. As a result, the function is injective in nature.

Formula of injective function

Consider two arbitrary items x and y in the domain of f.

Enter f(x) = f (y).

Solve the equation f(x) = f (y) If f(x) = f(y) results in x = y, then f : A ⟶ B is a one-one function or an injection.

If the function is supplied in the form of ordered pairs, and two ordered pairs do not have the same second element, then the function is one-one.

Conclusion

We conclude in this article that, an injective function also known as injection or one-to-one function. In mathematics is a function f that maps distinct elements to separate elements. They are important in determining an inverse function. For example, y=2x, which is both injective (since y1=y2) implies that (x1=x2). It is vital to note that no two elements in the domain map to the same codomain value. This is referred to as an injective function. An injective function is one in which no two domain items map to the same value in the codomain.