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Q.12 Letf : R\rightarrow R be defined asf(x) = 3x. Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

One - One : Let                                                               f is one-one.   Onto: We have  , then there exists     such that                . Hence, the function is one-one and onto. The correct answer is A .

Q.11 Let f : R \rightarrow R be defined as f(x) = x^4 . Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

One- one: For     then                                                                           does not imply that  example :     and     f is not  one- one  For    there is no x in R such that    f is not onto. Hence, f is neither one-one nor onto.  Option D is correct.

Q. 10 Let A = R - \{3\} and B = R - \{1\}. Consider the functionf : A\rightarrow B defined by
         f(x) = \left (\frac{x-2}{x-3} \right ) . Is f one-one and onto? Justify your answer.

Let  such that                                                                                                                                                                                                                                 f is one-one. Let,        then           such that                                                                                                    ...

Q.9

Let f : N \rightarrow N be defined by

                                f(n) = \left\{\begin{matrix} \frac{n+1}{2} & if\;n\;is\;odd \\ \frac{n}{2} & if\;n\;is\;even \end{matrix}\right.   for all n\in N.

State whether the function f is bijective. Justify your answer.

   ,        Here we can observe,             and        As we can see   but       f is not one-one. Let,    (N=co-domain) case1   n be even    For ,          then there is  such that  case2   n be odd For  ,    then there is  such that    f is onto. f is not one-one but onto hence, the function f is not bijective.      

Q. 8 Let A and B be sets. Show that f : A \times B \rightarrow B \times A such that f (a, b) = (b, a) is
bijective function.

Let  such that                                       and                        f is one- one Let,   then there exists   such that    f is onto. Hence, it is bijective.      

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

  Let  there  be    such that                                                                                                                                                                                              For      and    f is not one-one. Let there be    (-2 in codomain of R)                                              There does not exists any x in domain R  such that         ...

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

(i) f: R\rightarrow R defined by f(x) = 3 -4x

Let  there  be    such that                                                                                                                                                                                                  f is one-one. Let there be ,                                                                            Puting value of x,                                                  ...

Q.6 Let A = \{1, 2, 3\},B = \{4, 5, 6, 7\} and let f = \{(1, 4), (2, 5), (3, 6)\} be a function
from A to B. Show that f is one-one.

   Every element of A has a distant value in f. Hence, it is one-one.    

Q. 5 Show that the Signum Function f : R \rightarrow R, given by

                            f (x) = \left\{\begin{matrix} 1 & if\;x>0 \\ 0& if\;x=0 \\ -1& if\;x<0 \end{matrix}\right.

is neither one-one nor onto.

  is given by  As we can see       ,  but  So it is not one-one. Now, f(x) takes only 3 values (1,0,-1) for the element -3 in codomain  ,there does not exists x in domain  such that . So it is not onto. Hence, signum function is neither one-one nor onto.  

Q.4 Show that the Modulus Function f : R → R, given by f (x) = | x |, is neither one-one nor onto, where | x | is x, if x is positive or 0 and | x |  is - x,  if x is negative.

One- one: For      then                                                                                           f is not one- one i.e. not injective. For  , We know    is always positive there is no x in R such that    f is not onto i.e. not surjective. Hence,  , is neither one-one nor onto.

Q.3 Prove that the Greatest Integer Function f : R\longrightarrow R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

    One- one: For      then      and                                      but       f is not one- one i.e. not injective. For    there is no x in R such that    f is not onto i.e. not surjective. Hence, f is not injective but not surjective.

Q.2 Check the injectivity and surjectivity of the following functions:

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

 

One- one: For     then                                                                                               f is one- one i.e. injective. For    there is no x in Z such that    f is not onto i.e. not surjective. Hence, f is injective but not surjective.

Q.2 Check the injectivity and surjectivity of the following functions:

(iv) f: N \rightarrow N given by f(x) = x^3

One- one:   then                                                                           f is one- one i.e. injective. For  there is no x in N such that    f is not onto i.e. not surjective. Hence, f is injective but not surjective.

Q.2 Check the injectivity and surjectivity of the following functions:

(iii) f: R \rightarrow R given by f(x) = x^2

One- one: For     then                                                                          but      f is not one- one i.e. not injective. For    there is no x in R such that    f is not onto i.e. not surjective. Hence, f is not injective and not surjective.

Q.2 Check the injectivity and surjectivity of the following functions:

(ii) f : Z \rightarrow Z given by f(x) = x^2

One- one: For     then                                                                          but      f is not one- one i.e. not injective. For    there is no x in Z such that    f is not onto i.e. not surjective. Hence, f is neither injective nor surjective.

Q.2 Check the injectivity and surjectivity of the following functions:

(i) f : N\rightarrow N given by f(x) = x^2

 

One- one:   then                                                                           f is one- one i.e. injective. For  there is no x in N such that    f is not onto i.e. not surjective. Hence, f is injective but not surjective.  

Q.1 Show that the function f: R_* \longrightarrow R_{*} defined by f(x) = \frac{1}{x} is one-one and onto,
where R is the set of all non-zero real numbers. Is the result true, if the domain
R is replaced by N with co-domain being same as R?

Given,  is defined by  . One - One :                                                              f is one-one. Onto: We have  , then there exists      ( Here ) such that                . Hence, the function is one-one and onto. If the domain R∗ is replaced by N with co-domain being same as R∗   i.e.    defined by                                                                                 ...
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