Mathematics

# If $f(x)$ is odd function and $f(1)=a$, and $f(x+2)=f(x)+f(2)$ then the value of $f(3)$ is

$3a$

##### SOLUTION
Given that $f(x)$ is an odd function and $f(1)=a$.....(1) and

$f(x+2)=f(x)+f(2)$.....(2).

Since $f(x)$ is an odd function then we've, $f(-x)=-f(x)$ for all $x$.
Then we've, $f(-1)=-f(1)=-a$.......(3).

Now from (2) we get,

$f(-1+2)=f(-1)+f(2)$
or, $f(1)=f(-1)+f(2)$
or, $a=-a+f(2)$ [ Using (1) and (3)]
or, $f(2)=2a$......(4).

Now again from (2) we've,

$f(1+2)=f(1)+f(2)$
or, $f(3)=a+2a$ [ Using (1) and (4)]
or, $f(3)=3a$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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