Home
Class 12
MATHS
A function f from the set of natural ...

A function `f` from the set of natural numbers to the set of integers defined by `f(n)={(n-1)/2,\ w h e n\ n\ i s\ od d-n/2,\ w h e n\ n\ i s\ e v e n` (a) neither one-one nor onto (b) one-one but not onto (c) onto but not one-one (d) one-one and onto both

Text Solution

Verified by Experts

The correct Answer is:
bijective
When n is even, let
`f(2m_(1))=f(2m_(2))`
or `-(2m_(1))/(2)=-(2m_(2))/(2)`
or `m_(1)=m_(2)`
When n is odd, let
`f(2m_(1)+1)=f(2m_(2)+1)`

or `(2m_(1)+1-1)/(2)=(2m_(2)+1-1)/(2) " or "m_(1)=m_(2)`
Therefore, `f(x)` is one-one.
Also, when n is even, `-(n)/(2)=-(2m)/(2)= -m.`
When n is odd, `(n-1)/(2)=(2m+1-1)/(2)=m.`
Hence, the range of the function is Z.
Therefore, function is onto.
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.5|5 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.6|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.3|9 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE ENGLISH|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

A function f from the set of natural numbers to integers defined by f(n)={(n-1)/2,\ w h e n\ n\ i s\ od d-n/2,\ w h e n\ n\ i s\ e v e n is (a) neither one-one nor onto (b) one-one but not onto (c) onto but not one-one (d) one-one and onto both

A function f from the set of natural numbers to integers is defined by n when n is odd f(n) =3, when n is even Then f is (b) one-one but not onto a) neither one-one nor onto (c) onto but not one-one (d) one-one and onto both

The function f:[0,\ oo)->R given by f(x)=x/(x+1) is (a) one-one and onto (b) one-one but not onto (c) onto but not one-one (d) neither one-one nor onto

If f:[0,oo)->[0,oo) and f(x)=x/(1+x), then f(x) is (a) one-one and onto (b)one-one but not onto (c)onto but not one-one (d)neither on-one nor onto

Let f: R->R be a function defined by f(x)=(x^2-8)/(x^2+2) . Then, f is (a) one-one but not onto (b) one-one and onto (c) onto but not one-one (d) neither one-one nor onto

Let f: Z->Z be given by f(x)={x/2,\ if\ x\ i s\ e v e n,0,\ if\ x\ i s\ od d . Then, f is (a) onto but not one-one (b) one-one but not onto (c) one-one and onto (d) neither one-one nor onto

The function f : R rarr [ -1,1 ] defined by f(x) = cos x is (a) both one-one and onto (b) not one-one, but onto (c) one-one, but not onto (d) neither one-one, nor onto

Let M be the set of all 2xx2 matrices with entries from the set R of real numbers. Then the function f: M->R defined by f(A)=|A| for every A in M , is (a) one-one and onto (b) neither one-one nor onto (c) one-one but not onto (d) onto but not one-one

Let the function f: RrarrR be defined by f(x)=2x+sinx for x in Rdot Then f is (a)one-to-one and onto (b)one-to-one but not onto (c)onto but not-one-to-one (d)neither one-to-one nor onto

If f: (e,oo) rarr R & f(x)=log[log (logx)], then f is - (a)f is one-one and onto (b)f is one-one but onto (c)f is onto but not one-one (d)the range of f is equal to its domain