\large\sf\fbox\red{Question:-} check the injectivity and subjectivity of the function: \sf \large f:N \longmapsto \: N \: given \: by \: f(x) = x {}^(3) .​

Question

`\large\sf\fbox\red{Question:-}`

check the injectivity and subjectivity of the function:


`\sf \large f:N \longmapsto \: N \: given \: by \: f(x) = x {}^(3) .`​

Answer 1

This function is injective only.

Explanation:

N = natural numbers = positive whole numbers equal to or greater than 1.

Therefore
`N \rightarrow N` means that both the input and output are natural numbers.

A function is surjective if for every element in the codomain, there is at least one element in the domain.

Therefore, this function is NOT surjective. For example, there is no natural number for x for which f(x) = 2.

A function is injective if for every element in the codomain, there is at most one element in the domain. This is also called "One-to-One".

Therefore, this function IS injective as there is either none or only one value of x for each value in the codomain.

Answer 2

• The function is injective but nor surjective

Explanation:

We see that:

• f(x) = x³

For any x₁ and x₂ ∈ N,

• f(x) = x₁³ = x₂³ ⇒ x₁ = x₂, both are natural numbers

It it confirmed one-to-one, hence it is injective

Check the surjectivity:

f(x) = y ∈ N

• x³ = y
• y = ∛x

Let y = 2, then:

• 2 = x³
• x = ∛2 ∉ N

Since x is not natural, the function is not surjective