Into functions: Definition , examples - physicscatalyst's Blog

Into functions are those functions where range is a subset of the codomain of the functions but range is not equal to co-domain

Definition

A function f: A-> B is said to be into if every element of B is not image of some element of A under f, i,e
for every y ? B, there does exists a element x in A where $f(x) \ne y$

Now we know that A function f: A-> B is said to be onto(surjective) if every element of B is the image of some element of A under f.

So it means a function which is not onto is a into function

Examples

The Greatest integer function

f : R -> R f(x) = [x], rounds a real number x down to the nearest integer.

For example, f(3.8)=[3.8] = 3, f(-2.4)=[-2.4] = -3, and $f(\pi)=[\pi] = 3$

The graph of the function is

We can see that all the values of codomain does not have the preimage in domain

The absolute value function:

f: R -> R f(x) = |x|
For example, f(3) =|3| = 3, f(2)=|-2| = 2, and |0| = 0.

The graph of the function is

We can see that all the values of codomain does not have the preimage in domain

Square function

$f(x) =x^2$

for example $f(1) =1^2=1$ , $f(-1)= (-1)^2 = 1$

We can see that all the values of codomain does not have the preimage in domain

Difference between into functions and onto functions

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