Question

$\large\sf\fbox\red{Question:-}$
check the injectivity and subjectivity of the function:

$\sf \large f:N \longmapsto \: N \: given \: by \: f(x) = x {}^{3} .$
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Answer 1

Question:-

Check the injectivity and subjectivity of the function:

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

Given:-

The function

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

To Check:-

The injectivity and subjectivity of the given function.

Solution:-

The function

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

Clearly for x, y ∈ N,

f(x) = f(y)

⇒ x³ = y³

⇒ x = y

⇒ f is injective.

Now, let 2 ∈ N. But, we can see that there does not exist any x in N such that.

f(x) = x³ = 2.

⇒ f is not subjective.

Answer:-

Therefore, function f is injective but not subjective.

Hope you have satisfied. ⚘

Answer 2

Answer:

The function

f:N f(x) ⟼N given byf(x) = x³.

Clearly for x, y ∈ N,

f(x) = f(y)

⇒ x³ = y³

⇒ x = y

⇒ f is injective.

Now, let 2 ∈ N. But, we can see that there does not exist any x in N such that,

f(x) = x³ = 2.

⇒ f is not subjective.

Therefore, function f is injective but not subjective.

$thanks$