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/
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