Find the value of f(2)

1. May 10, 2013

utkarshakash

1. The problem statement, all variables and given/known data
If f(x) is a polynomial function satisfying 2+f(x)f(y)=f(x)+f(y)+f(xy), x,y belongs to R and if f(2)=5, then find the value of f(f(2))

2. Relevant equations

3. The attempt at a solution
The question clearly seeks the value of f(5). I put x=0 and y=2. Then
2+f(0)f(2)=f(0)+f(2)+f(0)
2+5f(0)=2f(0)+5
f(0)=1

Now I put x=0 and y=5
2+f(0)f(5)=f(0)+f(5)+f(0)
f(5)=f(5)

2. May 11, 2013

ehild

f(x) is a polynomial function. What does it mean?

ehild

3. May 11, 2013

haruspex

Consider a substitution f(x) = g(x) + c. Can you find a value of c that simplifies the equation?

4. May 11, 2013

utkarshakash

Nothing special I can think of.

5. May 11, 2013

Office_Shredder

Staff Emeritus
I'm going to second this suggestion

6. May 11, 2013

ehild

f(x) is a polynomial function, of form f(x)=a0+a1x+a2x2+a3x3+...

What relations do you get for the coefficients from the given equation and data?

ehild

Last edited: May 11, 2013
7. May 11, 2013

utkarshakash

I still don't know the degree of polynomial

8. May 11, 2013

utkarshakash

Ok following your method I arrive at this

2+g(x)g(y)+(c-1){g(x)+g(y)}=3c-c2+g(xy)

9. May 12, 2013

haruspex

Right, so what value of c will simplify that greatly?

10. May 12, 2013

ehild

That is you need to figure out. From the condition f(2)=5 you get a relation between f(y) and f(2y), and that can be fulfilled with polynomials of a certain degree.

ehild

Last edited: May 12, 2013
11. May 12, 2013

utkarshakash

The only number I can think of is 1

12. May 12, 2013

haruspex

Right, so what equation do you get for g()? When you have that, suppose α is a root of g(x). What other root(s) can you then deduce?

Last edited: May 12, 2013
13. May 12, 2013

ehild

I show my way as I think it is quite simple and straightforward.

2+f(x)f(y)=f(x)+f(y)+f(xy)

From f(2)=5 follows: 2+5f(x)=5+f(x)+f(2x) ----> -3+4f(x)=f(2x)*

f(x) is a polynomial f(x)=a0+a1x+a2x2+....+akxk+...

Plug into * and compare the coefficients of powers of x on both sides

-3+4(a0+a1x+a2x2+....+akxk+...)=a0+2a1x+4a2x2+....+2kakxk+...

-3+4a0=a0
4a1=2a1
4a2=4a2
.
.
.
4ak=2kak

What is the degree of the polynomial?

ehild

14. May 12, 2013

haruspex

Ah, but mine is so elegant

15. May 12, 2013

Dick

Elegant and cute, but less obvious.

16. May 12, 2013

ehild

I must be dumb but still do not know your solution.

ehild

17. May 12, 2013

Dick

I saw it. g(x)g(y)=g(xy) means if x is root of g then xy must be a root of g for ANY y. Severely limits the choice of roots. I think this is little too subtle.

18. May 12, 2013

ehild

I reached here, but what after? f(x)=1+xh(x). But it is obvious as f(x) is polynomial, and f(0)=1 (obtained by the OP already). Find the possible root of h?

ehild

Last edited: May 13, 2013
19. May 13, 2013

Office_Shredder

Staff Emeritus
I'm not sure what h is supposed to be here. But once you get to the post you quoted you say "ah, g(x) must be xk" and use f(2) = 5 to figure out what k is

20. May 13, 2013

ehild

I do not see that "ah" Perhaps I stick to my version too much which gives the degree at once.

ehild

Last edited: May 13, 2013