Home
Class 12
MATHS
A real-valued functin f(x) satisfies the...

A real-valued functin `f(x)` satisfies the functional equation `f(x-y)=f(x)f(y)-f(a-x)f(a+y),` where a given constant and `f(0)=1.` Then prove that `f(x)` is symmetrical about point (a, 0).

Text Solution

AI Generated Solution

To prove that the function \( f(x) \) is symmetrical about the point \( (a, 0) \), we will follow these steps: ### Step 1: Substitute \( x = 0 \) and \( y = 0 \) in the functional equation The functional equation is given by: \[ f(x - y) = f(x)f(y) - f(a - x)f(a + y) ...
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.15|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise Single Correct Answer Type|125 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.13|7 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 real valued function f(x) satisfies the functional equation f(x-y) = f(x) f(y) - f(a-x) f(a+y) , where a is a given constant and f(0)=1 , f(2a-x) =?

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

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

If f(x) is an even function, then the curve y=f(x) is symmetric about

A function f:R->R satisfies the relation f((x+y)/3)=1/3|f(x)+f(y)+f(0)| for all x,y in R. If f'(0) exists, prove that f'(x) exists for all x, in R.

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

Let f(x) be a function satisfying f(x+y)=f(x)+f(y) and f(x)=x g(x)"For all "x,y in R , where g(x) is continuous. Then,

If a function 'f' satisfies the relation f(x)f^('')(x)-f(x)f^(')(x) -f^(')(x)^(2)=0 AA x in R and f(0)=1=f^(')(0) . Then find f(x) .

Find function f(x) which is differentiable and satisfy the relation f(x+y)=f(x)+f(y)+(e^(x)-1)(e^(y)-1)AA x, y in R, and f'(0)=2.