Q.7 In each of the following cases, state whether the function is one-one, onto or
bijective. Justify your answer.

(ii) f : R\rightarrow R defined by f(x) = 1 + x^2

Answers (1)

f : R\rightarrow R

f(x) = 1 + x^2 

Let  there  be  (a,b) \in R  such that f(a)=f(b)

                                                          1+a^{2} = 1 +b^{2}

                                                               a^{2}=b^{2}

                                                                 a = \pm b

For   f(1)=f(-1)=2   and  1\neq -1

\therefore f is not one-one.

Let there be -2 \in R   (-2 in codomain of R)

                                   f(x) = 1 + x^2 = -2 

        There does not exists any x in domain R  such that  f(x) = -2                        

                           \therefore  f is not onto.

Hence, f is neither one-one nor onto.

   

 

 

 

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