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### Find Polynomial G(x) (Impossibility)

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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

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/
```
Associated Topics:
High School Calculus

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