A function f(x) is defined as f(x+y)=f(x)+f(y) for all real value of x

Question

A function f(x) is defined as  f(x+y)=f(x)+f(y)  for all real value of x and y, and  f(1)=3 . What is the value of  f(\frac{2}{3}) ?

A. 0
B.  \frac{2}{3} 
C. 1
D.  \frac{4}{3} 
E. 2

nick1816 · Accepted Answer

rajatchopra1994  Though you got correct answer, your approach is not correct.

f(\frac{1}{3}+\frac{1}{3})= f(\frac{1}{3})+f(\frac{1}{3})

f(\frac{2}{3})= 2f(\frac{1}{3}) ......(1)

also,  f(\frac{2}{3}+\frac{1}{3})= f(\frac{2}{3})+f(\frac{1}{3})

f(1)= 2f(\frac{1}{3})+f(\frac{1}{3})

f(1)= 3f(\frac{1}{3})

f(\frac{1}{3})= \frac{3}{3}=1

and   f(\frac{2}{3})= 2*1=2

rajatchopra1994 · Answer

IMO-E I am not sure. As Given F(1) = 3 We need for F(2/3), By Ratio Proportionate, its is 66.6% So , 66.6% of 3 is 2.

Please give Kudos, If you find my explanation Good Enough

allkagupta · Answer

Can somebody explain this

TestPrepUnlimited · Answer

We don't know any of the function values besides f(1) = 3 so we have to start from there. The target is to get the input value of  \frac{2}{3}  instead of 1. We can observe  f(x+x) = f(x) + f(x) , or  f(2x) = 2f(x) . This can be expanded to  f(3x) = 3f(x)  (from  f(x+y)=f(x)+f(y)  let y = 2x and plug in  f(2x) = 2f(x) ).

Therefore we can start from x = 1/3 and plug in  f(3x) = 3f(x)  to get  f(3 * \frac{1}{3}) = 3f(\frac{1}{3}) = 3 .

f(\frac{1}{3}) = 1

f(\frac{2}{3}) = 2f(\frac{1}{3}) = 2

Ans: E

BrushMyQuant · Answer

Given that  f(x+y)=f(x)+f(y)  and  f(1)=3  and we need to find the value of  f(\frac{2}{3})

\frac{2}{3}  =  \frac{1}{3}  +  \frac{1}{3}  
So, f( \frac{1}{3}  +  \frac{1}{3} ) = f( \frac{1}{3} ) + f( \frac{1}{3} )
=> f( \frac{2}{3} ) = 2*f( \frac{1}{3} )
=> f( \frac{1}{3} ) =  \frac{1}{2}  * f( \frac{2}{3} )

Now, 1 =  \frac{1}{3}  +  \frac{2}{3} 
 => f(1) = f( \frac{1}{3}  +  \frac{2}{3} ) = f( \frac{1}{3} ) + f( \frac{2}{3} ) =  \frac{1}{2}  * f( \frac{2}{3} ) + f( \frac{2}{3} ) 
=  \frac{3}{2}  * f( \frac{2}{3} )

And f(1) = 3 (given)
=>  \frac{3}{2}  * f( \frac{2}{3} ) = 3
=> f( \frac{2}{3} ) = 2

So,  Answer will be E 
Hope it helps!

Watch the following video to learn the Basics of Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

Dhwanii · Answer

nick1816 , is it allowed to use ratio proportion method for functions ?
2. How did you get f(1/3)=3/3=1 (2nd last step) 
Is it because, 
f(1)=3f(1/3)
3 = 3f(1/3) (as qs states f(1) = 3)
3/3 = f(1/3)
1= f(1/3)
If there's any other reason please correct me, thanks

Oppenheimer1945 · Answer

One example of this functional equation is y=ax; a=1/3, f(2/3)=2

bumpbot · Answer

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A function f(x) is defined as f(x+y)=f(x)+f(y) for all real value of x

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