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Q. 14 Let  be a function defined as . The inverse of  is the map given by

(A)

(B)

(C)

(D)

Let f inverse   Let y be element of range f. Then there is     such that                                                                                                                            Now , define     as                                                                                                                                                                                 ...

Q.  13 If  be given by , then  is

(A)

(B)

(C)

(D)

Thus, is x. Hence, option c is correct answer.

Q. 12 Let be an invertible function. Show that the inverse of  is , i.e.,

To prove:  Let    be a invertible function.   Then there is      such that     and      Also,                 and                              and          Hence,     is invertible function and f is inverse of . i.e.

11 Consider given by ,  and . Find and show that  .

,  and Let there be a function g such that   i.e.      ,        and       Now , we have And,          , here                    Hence, f exists and  is g.       ,       and    Let inverse of  be h such that   And        ,  here   Thus,    It is noted that h=f. Hence,.

Q.10 Let be an invertible function. Show that f has a unique inverse.
(Hint: suppose and  are two inverses of . Then for all ,
. Use one-one ness of f).

Let be an invertible function Also, suppose f has two inverse  For , we have                                                  [f is invertible implies f is one - one]                                              [g is one-one] Thus,f has a unique inverse.

Q .9 Consider given by . Show that  is invertible with

One- one: Let                                                                                   Since, x and y are positive.                f is one-one. Onto: Let for    ,                                                                                                                                                                                                                          ...

Q. 8 Consider  given by . Show that  is invertible with the
inverse of  given by , where is the set of all non-negative
real numbers.

One- one: Let                                                                 f is one-one. Onto: Let for    ,                                                                                                                 for  there is   such that                                             f is onto. Since, f is one-one and onto so it is invertible. Let      by   Now,                     ...

Q. 7  Consider given by . Show that f is invertible. Find the
inverse of .

is given by   One-one : Let                                    f is one-one function. Onto: So, for  there is     ,such that                                                                               f  is onto. Thus, f is one-one and onto so  exists. Let,  by  Now,                                                                                                                        ...

Q. 6 Show that , given by is one-one. Find the inverse of the function

One -one:                                                                f is one-one. It is clear that   is onto. Thus,f is one-one and onto so inverse of f exists. Let g be inverse function of f in   let y be an arbitrary element of range f Since,  is onto ,so           for                                            ,

Q. 3 State with reason whether following functions have inverse

(iii)    with

(iii)    with         From the definition, we can see the set  have distant values under h.  h is one-one . For every element y of set ,there exists an element x  in  such that    h is onto Thus, h is one-one and onto so h has an inverse function.

Q. 5 State with reason whether following functions have inverse

(ii)   with

(ii)   with       From the definition, we can conclude :  g is not one-one. Hence, function g does not have inverse function.

Q5.State with reason whether following functions have inverse

(i)  with

(i)  with     From the given definition,we have:  f is not one-one. Hence, f do not have inverse function.

Q.4  If  show that ,  for all . What is the inverse of ?

,  for all   Hence,the given function is invertible and the inverse of  is  itself.

Q.3 Find gof and fog, if

(ii) and

The solution is as follows (ii)  and

Q.3 Find  and , if

(i)   and

and

Q.2 Let , and  be functions from  to . Show that

To prove :                                                                                                                   Hence,      To prove:                                                                                                                                      Hence,

Q.1 Let and be given by
and . Write down .

Given :                    and                                   and                                                                  Hence,    =
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