Find Polynomial G(x) (Impossibility)

Date: 7/1/96 at 1:11:10
From: Anonymous
Subject: Find Polynomial G(x)

Dear Dr. Math, 

	For all real X, F(x) = x+3 and G(x) is a polynomial of degree 2 
such that G(F(x)) = x+2. Find G(x).

	My first problem is that I don't really understand functions. The 
definition that I have is that for F(x) = x+3, then F(y) = y+3 and y+3 
would replace everything in F(x) where x was written. If that is true, 
I have the following:

G(X) = X^2 + 2X + 9
G(F(X)) = (X+3)^2 + 2(X+3) + 9 AND G(X) = X+2

I am stuck big time. Please help if you can.
Thank you again,
	Nancy Geldermann

Date: 7/1/96 at 6:20:38
From: Doctor Anthony
Subject: Re: Find Polynomial G(x)

We have f(x) = x+3 and g(x) = ax^2+bx+c  
(i.e. a polynomial of degree 2)

We are also given that g(f(x)) = x+2, so we have the following 
identity:

  a(x+3)^2 + b(x+3) + c == x+2   (where == means identically equal 
to.)

This is true for all values of x, so if x = -3 we get

       c = -3 + 2 = -1

We can also equate the coefficients of equal powers of x on the two 
sides of the identity, and since coefficient of x^2 is 0 on the right 
hand side, it follows that coefficient on the left must be 0.  This 
gives a = 0.  The identity now can be written

         b(x+3) - 1 == x + 2    and equating coefficients of x we see 
that b=1. 

      So we have a = 0,    b = 1,     c = -1

Thus g(x) = x-1.  In view of this result, I am surprised the question 
said that g(x) was a polynomial of degree 2.  Let us check whether 
g(f(x)) = x+2

  g(x+3) = (x+3) - 1
         =  x+3  - 1
         =  x+2      So the answer is correct.  g(x) = x-1

Could you check that your expression for g(f(x)) = x+2 is as given in 
the question.  As you can see this makes it impossible for g(x) to be 
a polynomial of degree 2.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/