Understanding Function Properties: Solving f(x) Equations for Homework

[Arman777]

Homework Statement

$f(xy)=f(x)+f(y)$ and $f(16)=16$, $f(2)=?$

Homework Equations

function properities

The Attempt at a Solution

$f(x)=y$ so
$f(xf(x))=f(x)+f(f(x))$
then I put
$xf(x)=16$
and
$f(x)+f(f(x))=16$
but this not much make sense,
I am really stucked

 

[PeroK]

What about $f(4)$?

 

[Arman777]

What about $f(4)$?

I don't know,where should I put it ?

 

[PeroK]

I don't know,where should I put it ?

Think about it!

 

[Arman777]

Think about it!

I am thinking,I didnt understand where it will lead. $f(4f(4))=f(4)+f(f(4))$ so ?

 

[PeroK]

I am thinking,I didnt understand where it will lead. $f(4f(4))=f(4)+f(f(4))$ so ?

Why are you taking $f(f(x))$?

 

[Arman777]

Why are you taking $f(f(x))$?

what's that mean ?

 

[pasmith]

f(16) = f(2*8) = f(2) + f(8) ...

 

[Arman777]

f(16) = f(2*8) = f(2) + f(8) ...

I found that but ıts not going further.And from $xf(x)=16$ and from $f(x)+f(f(x))=16$ you said , $f(x)+f(16/x)=16$
so $f(4)=8$ and $f(2)+f(8)=16$ but we can't find $f(8)$ and also,I think we can't do these steps cause, we said $f(4)=8$ but we know that
$xf(x)=16$ so $4f(4)=32≠16$.

simply $f(x)+f(16/x)=16$ this is only true for an special $x$ not all $"x"$ s

 

[SammyS]

Homework Statement

$f(xy)=f(x)+f(y)$ and $f(16)=16$, $f(2)=?$

You should NOT be assuming that $\ y = f(x)\$ in the above statement.

The author of this question might just as well have stated this as follows:

$f(ab)=f(a)+f(b)$​

Aa examples,consider the following:

$f(35)=f(5)+f(7)$

$f(9)=f(3)+f(3)$

$=2f(3)$​

Homework Equations

function properties

The Attempt at a Solution

$f(x)=y$ so
$f(xf(x))=f(x)+f(f(x))$
then I put
$xf(x)=16$
and
$f(x)+f(f(x))=16$
but this not much make sense,
I am really stuck ed

 

[Arman777]

$f(16)=f(2)+f(8)=16$
$f(8)=f(4)+f(2)$
and $f(4)=8$ so,
$f(8)-f(2)=8$
$f(8)+f(2)=16$
$f(2)=4$

 

[Arman777]

Still awkard..there must a solution which we can assume $y=f(x)$ and then solve it

 

[SammyS]

Still awkard..there must a solution which we can assume $y=f(x)$ and then solve it

You should not assume that for the statement of this problem.

 

[Arman777]

You should not assume that for the statement of this problem.

Ok,thanks..I am trying to solve this for like hours by using $y=f(x)$ ,which I was wrong

 

[PeroK]

Ok,thanks..I am trying to solve this for like hours by using $y=f(x)$ ,which I was wrong

It's difficult to help any more without giving it away. Anyway, here's a bit hint: $4 \times 4 = 16$.

 

[Arman777]

It's difficult to help any more without giving it away. Anyway, here's a bit hint: $4 \times 4 = 16$.

Thanks

 

[PeroK]

?

$f(4 \times 4) = f(16)$

 

[Arman777]

$f(4 \times 4) = f(16)$

oh I see yeah..Thanks again

 

[SammyS]

oh I see yeah..Thanks again

I think @PeroK was trying to get you to answer his previous question: "What is ƒ(4) ?"

 

[Arman777]

I think @PeroK was trying to get you to answer his previous question: "What is ƒ(4) ?"

İts 8 I found it ?

 

[SammyS]

İts 8 I found it ?

There is a more direct path to finding f(4):

ƒ(16) = ƒ(4⋅4) = ƒ(4) + ƒ(4) = 2⋅ƒ(4)

ƒ(4) = ƒ(16)/2 = 8

You might similarly consider how ƒ(x²) is related to ƒ(x), etc.