#### Need solution for RD Sharma maths class 12 chapter Functions exercise 2.1 question 5 sub question (iv)

Injective but not surjective.

Given:

$f:Z \rightarrow Z$ given by$f(x)=x^3$

Hint:

One-one function means every element in the domain has a distinct image in the  co-domain.

Onto function means every element in the co-domain has at least one  pre image in the domain of function.

For any function to be a bijective, the given function be one-one and onto.

Solution:

Let us check for the given function is injection, surjection and bijection.

Let x and y be any two elements in the domain (N) such that$f(x)=f(y)$

\begin{aligned} &f(x)=x^{3}, f(y)=y^{3} \\ &f(x)=f(y) \\ &x^{3}=y^{3} \\ &x=y \end{aligned}

So,f is an injection

Surjection test:

Let y be any element in the codomain (N) , such that$f(x)=y$  for some element x in N .

$f(x)=y$

$x^3=y$

$x=\sqrt[3]{y}$ which may not be in N

So,f is not a surjection.

Here f is injective but not surjective, so the f is not a bijection.

The condition of bijection is as we know the function should be one-one and onto.

Hence for this, f is not bijection.