Home
Class 12
MATHS
Prove that f(x)gi v e nb yf(x+y)=f(x)+f(...

Prove that `f(x)gi v e nb yf(x+y)=f(x)+f(y)AAx in R` is an odd function.

Text Solution

AI Generated Solution

To prove that the function \( f(x) \) defined by the equation \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \) is an odd function, we need to show that \( f(-x) = -f(x) \) for all \( x \in \mathbb{R} \). ### Step-by-Step Solution: **Step 1: Understanding the Function Property** We start with the given property of the function: \[ ...
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise Solved Examples|15 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.1|15 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

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

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

Which of the following functions have the graph symmetrical about the origin? (a) f(x) given by f(x)+f(y)=f((x+y)/(1-x y)) (b) f(x) given by f(x)+f(y)=f(xsqrt(1-y^2)+ysqrt(1-x^2)) (c) f(x) given by f(x+y)=f(x)+f(y)AAx , y in R (d) none of these

Suppose the function f(x) satisfies the relation f(x+y^3)=f(x)+f(y^3)dotAAx ,y in R and is differentiable for all xdot Statement 1: If f^(prime)(2)=a ,t h e nf^(prime)(-2)=a Statement 2: f(x) is an odd function.

If the function f satisfies the relation f(x+y)+f(x-y)=2f(x),f(y)AAx , y in Ra n df(0)!=0 , then f(x) is an even fu n c t ion f(x) is an odd fu n c t ion If f(2)=a ,t h e nf(-2)=a If f(4)=b ,t h e nf(-4)=-b

Let f be a one-one function such that f(x).f(y) + 2 = f(x) + f(y) + f(xy), AA x, y in R - {0} and f(0) = 1, f'(1) = 2 . Prove that 3(int f(x) dx) - x(f(x) + 2) is constant.

Let f(x) be real valued and differentiable function on R such that f(x+y)=(f(x)+f(y))/(1-f(x)dotf(y)) f(x) is Odd function Even function Odd and even function simultaneously Neither even nor odd

Let a real valued function f satisfy f(x + y) = f(x)f(y)AA x, y in R and f(0)!=0 Then g(x)=f(x)/(1+[f(x)]^2) is

Let f:R to R be defined by f(x) =e^(x)-e^(-x). Prove that f(x) is invertible. Also find the inverse function.