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  • If f(x) satisfies the relation  f(x + y) = f(x) + f(y) ∀ x, y ∈ R. Then f(x) isOption: 1 Odd Function     &

If f(x) satisfies the relation  f(x + y) = f(x) + f(y) ∀ x, y ∈ R. Then f(x) is

Option: 1

Odd Function        


Option: 2

Even function


Option: 3

Both (1) and (2)


Option: 4

None


Answers (1)

best_answer

Even function:

If for a function f(x), f(-x) = f(x) then the function is known as even function. Even functions are symmetric about the y axis.

 

         

            y = x2                                            y = |x|                                              y = cos(x)

 

Odd function:

If for a function f(x), f(-x) = - f(x) then the function is known as odd function. Odd functions are symmetric about the origin.

              

                                y = sin(x)                                                           y = x3

 

NOTE: We can have functions that are neither even nor odd. Eg, y = x+1

 

 

Given, f(x + y) = f(x) + f(y)

Put x = 0, y = 0 

f(0 + 0) = f(0) + f(0) ⇒ f(0) = 0

Now, replace y with (-x)

f(x - x) = f(x) + f(-x)

f(0) =   f(x) + f(-x)

f(x) = -f(-x)

f(x) is odd function

Posted by

Kshitij

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