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Q.12 Let$f : R\rightarrow R$ be defined as$f(x) = 3x$. 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 function$f : A\rightarrow B$ 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 $n\in N$.

,        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

(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

(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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