Types of Functions

A function is a rule that assigns each element of one set to exactly one element of another set. In this topic, the main goal is to understand how different functions behave based on the way elements of the domain are mapped to elements of the codomain.

1. One-one (Injective) function

A function is called one-one if different elements of the domain have different images in the codomain.

2. Many-one function

A function is called many-one if two or more different elements of the domain have the same image in the codomain.

3. Onto function

A function is called onto if every element of the codomain has at least one pre-image in the domain.

4. Into function

A function is called into if at least one element of the codomain is not the image of any element of the domain.

5.Bijection

A function is called bijective if it is both one-one and onto

Show that the function f: N → N defined by f(x) = 2x is one-one but not onto.

Solution:

The function f is one-one.

Reason : Let x₁, x₂ ∈ N be such that f(x₁) = f(x₂) ⇒ 2x₁ = 2x₂ ⇒ x₁ = x₂

Thus, f(x₁) = f(x₂) ⇒ x₁ = x₂ ⇒ f is one-one.

The function f is not onto.

Reason: As 1 ∈ N (codomain of f) and there does not exist any x ∈ N (domain of f) such that f(x) = 1. So f is not onto.

Show that the function f: Z → Z defined by f(x) = x² + x, for all x ∈ Z, is a many-one function.

Solution:
Let x, y ∈ Z such that f(x) = f(y). Then f(x) = f(y)
⇒ x² + x = y² + y

⇒ (x² − y²) + (x − y) = 0

⇒ (x − y)(x + y + 1) = 0

⇒ x = y or x = −y − 1

Since f(x) = f(y) does not yield the unique solution x = y but also provides the solution x = −y − 1, so it is not a one-one function. For example, if y = 1, then x = 1 from x = y and also x = −2 from y = −x − 1. This means that 1 and −2 have the same image in B under f.

Hence, f is a many-one function.

State whether the function f: N → N given by f(x) = 5x is injective, surjective or both.

Solution:
For a function to be injective, it should be one-one and for surjective, it should be onto.

Given f(x) = 5x.

Let x₁, x₂ ∈ N. Then

f(x₁) = f(x₂) ⇒ 5x₁ = 5x₂ ⇒ x₁ = x₂, ∀ x₁, x₂ ∈ N

Hence, f(x) is an injective function.

Now, the range is a set comprising multiples of 5 ∀x in the domain. But codomain = N. Thus, the range ≠ codomain, or we can say that every element of the codomain does not have a pre-image in the domain.

∴ f(x) is not a surjection.

Note:
If the given function f were f: R → R, i.e., the domain and codomain both equal to R, then the function f(x) = 5x would have been a surjection.

Let X = {−1, 2, 5, 8, 11}, Y = {4, 6, 8} and f be the function from X to Y depicted by the adjoining arrow diagram, then f is

1. a many-one function.
   Reason: different elements 2, 5 and 8 of X have the same image 4
2. an into function.
   Reason: there exists 6 ∈ Y which is not the image of any element of X.
   

Thus, the function f from X to Y is a many-one into function.

State whether the function f: R → R defined by f(x) = 1 + x² is one-one, onto or bijective. 

Solution: Given, function f: R → R such that f(x) = 1 + x².
Let A and B be two sets of real numbers.

Let x₁, x₂ ∈ A such that f(x₁) = f(x₂).

⇒ 1 + x₁² = 1 + x₂² ⇒ x₁² − x₂² = 0 ⇒ (x₁ − x₂)(x₁ + x₂) = 0
⇒ x₁ = ± x₂. Thus f(x₁) = f(x₂) does not imply that x₁ = x₂.

For instance, f(1) = f(−1) = 2, i.e., two elements (1, −1) of A have the same image in B. So, f is a many-one function.

Now, y = 1 + x² ⇒ x = ±\[\sqrt{y-1}\]⇒ elements < y have no pre-image in A (for instance, an element −2 in the codomain has no pre-image in the domain A). So, f is not onto.

Hence, f is neither one-one nor onto. So, it is not bijective.

• Roll number to student: Each roll number identifies exactly one student, so this resembles a one-one idea when no roll number is repeated.

• Students to favourite subject: Many students may choose the same favourite subject, so this resembles a many-one function.

• Seats filled in a classroom: If every seat is occupied by one student, the situation resembles onto coverage; if some seats are empty, it resembles into.

• Password or code matching system: A perfect one-to-one matching between users and unique IDs is similar to a bijection.