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Provide solution for RD Sharma maths class12 Chapter Functions exercise 2.1 question 5 sub question (xiii)

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f:Q\rightarrow Q  defined by  f(x)=x^3+1


Injective (one-one): function means every element in the domain has a distinct image in the  co-domain.

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


Injection: Let x,y be any two element in the domain Q,

such that f(x)=f(y).    

\begin{aligned} &x^{3}+1=y^{3}+1 \\ &x^{3}=y^{3} \\ &x=y \end{aligned}

Therefore, f  is an injection.

Surjection: Let y  be any two element in the co-domain Q, such that f(x)=y  for some element x  in Q.
\begin{aligned} &f(x)=y \\ &x^{3}+1=y \\ &x^{3}=y-1 \\ &x=\sqrt[3]{(y-1)} \end{aligned} which may not be Q.

So, f  is not surjective.

Therefore, f is not  bijective.

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