Home
Class 12
MATHS
Let a function f:RtoR satisfy the equati...

Let a function `f:RtoR` satisfy the equation `f(x+y)=f(x)=f(Y)AAx, yepsilonR`. If the function f(x) is continuous at x=0, then

A

a periodic function

B

an even function

C

an odd function

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
We know that the function f(x) satisfying the property
`f(x+y)=f(x)+f(y)` for all ` x, y in R` has the formula
`f(x)=xf(1)` for all ` x in R`.
Clearly , it is an odd function.
Promotional Banner

Topper's Solved these Questions

  • REAL FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section I - Solved Mcqs|95 Videos

  • REAL FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|8 Videos

  • PROPERTIES OF TRIANGLES AND CIRCLES CONNECTED WITH THEM

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|55 Videos

  • SCALAR AND VECTOR PRODUCTS OF THREE VECTORS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|63 Videos

Similar Questions

Explore conceptually related problems

If a real valued function f(x) satisfies the equation f(x +y)=f(x)+f (y) for all x,y in R then f(x) is

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

A function f : R -> R^+ satisfies f(x+y)= f(x) f(y) AA x in R If f'(0)=2 then f'(x)=

A function f(x) satisfies the following property: f(xdoty)=f(x)f(y)dot Show that the function f(x) is continuous for all values of x if it is continuous at x=1.

If the function / satisfies the relation f(x+y)+f(x-y)=2f(x),f(y)AAx , y in R and f(0)!=0 , then

A function f: R -> R satisfy the equation f (x)f(y) - f (xy)= x+y for all x, y in R and f(y) > 0 , then

A function f : R→R satisfies the equation f(x)f(y) - f(xy) = x + y ∀ x, y ∈ R and f (1)>0 , then

A function f(x) satisfies the relation f(x+y) = f(x) + f(y) + xy(x+y), AA x, y in R . If f'(0) = - 1, then

The function f(x) satisfying the equation f^2 (x) + 4 f'(x) f(x) + (f'(x))^2 = 0

OBJECTIVE RD SHARMA ENGLISH-REAL FUNCTIONS -Chapter Test
  1. Let a function f:RtoR satisfy the equation f(x+y)=f(x)=f(Y)AAx, yepsil...

    Text Solution

    |

  2. The period of the function f(x)=sin^(4)3x+cos^(4)3x, is

    Text Solution

    |

  3. The value of integer n for which the function f(x)=(sinx)/(sin(x / n)...

    Text Solution

    |

  4. The period of the function f(x)=sin((2x+3)/(6pi)), is

    Text Solution

    |

  5. The domain of the function f(x)=sqrt(log((1)/(|sinx|)))

    Text Solution

    |

  6. The domain of the function f(x)=log(10) (sqrt(x-4)+sqrt(6-x)) is :

    Text Solution

    |

  7. Let f(x)=(sqrt(sinx))/(1+(sinx)^(1/3)) then domain f contains

    Text Solution

    |

  8. If f : R -> R is defined by f(x) = [2x] - 2[x] for x in R, where [x] i...

    Text Solution

    |

  9. If N denotes the set of all positive integers and if f : N -> N is def...

    Text Solution

    |

  10. The set of value of a for which the function f(x)=sinx+[(x^(2))/(a)] d...

    Text Solution

    |

  11. If f(x)={{:(-1, x lt 0),(0, x=0 and g(x)=x(1-x^(2))", then"),(1, x gt ...

    Text Solution

    |

  12. Find the equivalent definition of f(x)=max.{x^(2),(1-x)^(2),2x(1-x)...

    Text Solution

    |

  13. If f(x) is defined on [0,1], then the domain of f(3x^(2)) , is

    Text Solution

    |

  14. The function f(x) is defined in [0,1] . Find the domain of f(t a nx)do...

    Text Solution

    |

  15. The domain of definition of the real function f(x)=sqrt(log(12)x^(2)) ...

    Text Solution

    |

  16. The values of ba n dc for which the identity of f(x+1)-f(x)=8x+3 is sa...

    Text Solution

    |

  17. The function f(x)=sin""(pix)/(2)+2 cos ""(pix)/(3)-tan""(pix)/(4) is p...

    Text Solution

    |

  18. The period of the function sin""((pix)/(2))+cos((pix)/(2)), is

    Text Solution

    |

  19. If x in R, then f(x)=sin^(-1)((2x)/(1+x^(2))) is equal to

    Text Solution

    |

  20. If x in R , then f(x)=cos^(-1)((1-x^(2))/(1+x^(2))) is equal to

    Text Solution

    |

  21. The equivalent definition of the function f(x)=lim(n to oo)(x^(n)-x^(-...

    Text Solution

    |