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Show that the function
(i) f: N \rightarrow N: f(x)=x^{3} is one – one into
(ii) f: Z \rightarrow Z: f(x)=x^{3} is one – one into
CBSE | Class 12 | Exercise 2A | Functions | Maths | RS Aggarwal
Show that the function
(i) f: N \rightarrow N: f(x)=x^{3} is one – one into
(ii) f: Z \rightarrow Z: f(x)=x^{3} is one – one into

Solution:

(i)

f: N \rightarrow N: f(x)=x^{3} is one – one into.
f(x)=x^{3}
As the function f(x) is monotonically increasing from the domain N \rightarrow N \therefore f(x) is one -one
Range of f(x)=(-\infty, \infty) \neq N (codomain)
\therefore f(x) is into
\therefore f: \mathbb{N} \rightarrow \mathrm{N}: f(x)=x^{2} is one – one into.

(ii)

f: Z \rightarrow Z: f(x)=x^{3} is one – one into
f(x)=x^{3}
As the function f(x) is monotonically increasing from the domain Z \rightarrow Z \therefore f(x) is one -one
Range of f(x)=(-\infty, \infty) \neq Z (codomain)
\therefore f(x) is into
\therefore f: Z \rightarrow Z: f(x)=x^{3} is one - one into.

i

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