Home
Class 12
MATHS
Let f: A->A be a function such that fof=...

Let `f: A->A` be a function such that `fof=f` . Show that `f` is onto if and only if `f` is one-one. Describe `f` in this case.

Text Solution

Verified by Experts

Since A be a finite set and fof=f
So,
f:A→ A is onto that means every element in Y
set has a preimage in X set
That also means that every element in
X must have only one image in Y because both
X,Y are equal so the function is one - one
Promotional Banner

Topper's Solved these Questions

  • DIRECTION COSINES AND DIRECTION RATIOS

    RD SHARMA|Exercise Solved Examples And Exercises|67 Videos

  • HIGHER ORDER DERIVATIVES

    RD SHARMA|Exercise Solved Examples And Exercises|176 Videos

Similar Questions

Explore conceptually related problems

Show that the function f(x)=3x+2 is one one and onto

Let f:R to R where f(x) =sin x. Show that f is into. Also find the codomain if f is onto.

Using the definition, Prove that the function f:A to B is invertible if and only if f is both one-one and onto.

Show that the function f:N rarr N given by f(x)=3x is one one but not onto

If f is a function from RrarrR such that f(x)=x^(2)AA x inR , then show that 'f' is not one-one.

Let f: R-{n}->R be a function defined by f(x)=(x-m)/(x-n) such that m!=n 1) f is one one into function2) f is one one onto function3) f is many one into funciton4) f is many one onto funcionn then

Show that the function f:N rarr N, given by f(x)=2x, is one-one but not onto.

Let f:A rarr B and g:B rarr C be two functions.Then; if gof is onto then g is onto; if gof is one one then f is one-one and if gof is onto and g is one one then f is onto and if gof is one one and f is onto then g is one one.

Show that the function f:R rarr R:f(x)=sin x is neither one-one nor onto

If functions f:A to B and g : B to A satisfy gof= I_(A), then show that f is one-one and g is onto.

RD SHARMA-FUNCTION-Solved Examples And Exercises
  1. Let f: ZvecZ be defined by f(n)=3n for all n in Z and g: Zvec be defi...

    Text Solution

    |

  2. Let f: RvecR be a function given by f(x)=a x+b for all x in R . Find ...

    Text Solution

    |

  3. Let f: A->A be a function such that fof=f . Show that f is onto if and...

    Text Solution

    |

  4. Let f,g: RvecR be two functions defined as f(x)=|x|+x and g(x)=|x|-x ,...

    Text Solution

    |

  5. Find gof and gof when f: R->R and g: R->R is defined by f(x)=2x+3 and ...

    Text Solution

    |

  6. Find gof and gof when f: R->R and g: R->R is defined by f(x)=2x+x^2 an...

    Text Solution

    |

  7. Find fog(2) and gof(1) when: f: R rarr R ;f(x)=x^2+8 and g: R rarr ...

    Text Solution

    |

  8. Find gof and fog when f: R->R and g: R->R is defined by f(x)=x and g(x...

    Text Solution

    |

  9. Find gof and fog when f: R->R and g: R->R is defined by f(x)=x^2+2x-3 ...

    Text Solution

    |

  10. Find gof and gof when f: R->R and g: R->R is defined by f(x)=8x^3 and ...

    Text Solution

    |

  11. Let f={(3,\ 1),\ (9,\ 3),\ (12 ,\ 4)} and ={(1,\ 3),\ (3,\ 3),\ (4,\ 9...

    Text Solution

    |

  12. Let f={(1,-1),(4,-2),(9,-3),(16 ,4)} and g={(-1,-2),(02 ,-4),(-3,-6),(...

    Text Solution

    |

  13. Let A={a ,\ b ,\ c} , B={u\ v ,\ w} and let f and g be two functions f...

    Text Solution

    |

  14. Find fog\ (2) and gof\ (1) when: f: R->R ; f(x)=x^2+8 and g: R->R ; g(...

    Text Solution

    |

  15. Let R^+ be the set of all non-negative real numbers. If f: R^+\ ->R...

    Text Solution

    |

  16. Let f: R->R and g: R->R be defined by f(x)=x^2 and g(x)=x+1 . Show tha...

    Text Solution

    |

  17. Let f: RvecR and g: RvecR be defined by f(x)=x+1 and g(x)=x-1. Show...

    Text Solution

    |

  18. Verify assoiativity for the following three mappings : f: NvecZ0...

    Text Solution

    |

  19. Consider f : N ->N, g : N ->Nand h : N ->Rdefined asf (x) = 2x, g (y) ...

    Text Solution

    |

  20. Given examples of two functions f:" "N ->N" "a n d""""""g:" "N->N such...

    Text Solution

    |