1why you don't want to try?
let F(x) and G(x) be two cubic polynomial functions such that F(G(x))=G(F(x)).
F(0)= -24,
G(0)=30
find the value of F(3) + G(6) ??
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17 Answers
1As far as solution says>>>>>>>>>
note that the polynomials f(x)=ax3 and g(x)=-ax3 commute under composition.Let h(x) =x+b be a linear polynomial.
and its inverse h-1(x)=x-b is also linear polynomial.
The composite polynomials h-1 f h and h-1 g h also commute ,since function composition is associative, and these polynomials are also cubic. solving for a and b. such that h-1 f h(0)=-24 , and h-1 g h(0) =30,
so ab3 -b=-24
and -ab3-b=30
so ,f(x)=(x-3)3 +3
and g(x)=-(x-3)3 -3
please explain whats been done here!!!
1@sharadapa..
For eg,
if f(x)= x+1, and g(x)=x+5;
then f(g(x))=x+6=g(f(x))
then we say that f(x) and g(x) COMMUTE;
similar thing happens for higher degree polynomials
1hey theprophet,can u plz tell me what are commuting polynomials and chebyshev polynomials?
341I too am intrigued. Because as far as I know commuting polynomials fall essentially under three categories.
(1) Chebyshev Polynomials
(2) P(x) = xn for some natural number
(3) Iterates of the same polynomial i.e. Pn(t) = P(P(...(P)...))(t) where P is composed n times with itself.
But here both are cubic and so I am wondering if I have missed something
1its a bit long,
the solution deals with some commution of polynomial
think!!
i ll post it later,
1...all i could conclude was that
1.f(x) has constant term -24 and g(x) has it as 30.
from given
F(0)= -24,
G(0)=30
all i could find was:[doubtful!]
2.coefficient of linear term of f(x) and g(x) is 1
3.leading coefficients of both f(x) and g(x) should be of opposite signs.
not my piece of cake[3]
1there are infinite such polynomials,
but in this case there is a unique one
inverse wouldn't help,
anyone trying?
1sir if u could plz post some of ur solving steps it would be quite a help
1hehe....i could not even trace the answer.....incorrect hone ka sawal hi nahin hai :P :P :P