Question

If the function f : N → N be defined by f(n) = 3n + 4, ∀ n ∈ N, then f is:

1. Surjective.
2. Injective.
3. Bijective.
4. Both 1 and 2
5. None of these

Answer 1

Correct Answer - Option 2 : Injective.

Concept:

• One-To-One (Injective) Function: The function f : A → B is one-to-one if every element of the range B corresponds to exactly one element of the domain A. i.e. If a ∈ A, b ∈ B and f(a) = f(b), then a = b.
• Onto (Surjective) Function: The function f : A → B is said to be onto, if every element of B is the image of some element of A. i.e. For every b ∈ B there exists an element a ∈ A such that f(a) = b.
• One-To-One and Onto (Bijective) Function: A function f : A → B is said to be one-one and onto, if it is both one-one and onto.

 

Calculation:

Let's check the given function f(n) = 3n + 4, ∀ n ∈ N for one-to-one and onto:

Injective: Let's say 3n + 4 = k \(\rm \Rightarrow n=\dfrac{k-4}{3}\). It means that for every value of k, we will get only a single value of n, therefore the function is injective.

Surjective: Since \(\rm n=\dfrac{k-4}{3}\), not all values of k will give n ∈ N. So, the function is not surjective.

Bijective: Since the function is injective but not surjective, it is not bijective.

So, the function f is only injective.