Question

Conceptual Problem

Answer 1

\(\mathsf{\textbf{Question : }}\)Let \(f(x)=x^2+2x\) and \(g(x)=x^3\), how many different roots does \(f\circ g(x)=g\circ f(x)\) have?

\(\textbf{Answer : }\) We are in a very nice situation here because \(g(x)\) is invertible, meaning \(\sqrt[3]{x}\) is uniquely defined for any real number \(x\). So we can rewrite this as \(g^{-1}\circ f\circ g(x)=f(x)\), which is true if and only if \(f(x)=x\)(since both \(g\) and \(g^{-1}\) are one-to-one). We solve \(f(x)=x\) for \(x\) by subtracting \(x\) from both sides to get \(x^2+x=0\). This can be factored as \(x(x+1)=0\), so our two solutions are \(x=0\) and \(x=-1\).\[\]
\(------------------------------------\)

I need to understand the answer, the "... which is true if and only if \(f(x)=x\)(since both \(g\) and \(g^{-1}\) are one-to-one)..." part.

Answer 2

wtf is \(f\circ g(x)\)? is \(\circ\) multiplication?

Answer 3

No, it's for composite function.

\[f \circ g(x) = f\left( g(x)\right)\]

Answer 4

oh ok i know that one but never seen it expressed with \(\circ\)

Answer 5

Oh, hmm this question was asked on math.stackexchange. I copied the whole LaTeX and question from there, and pasted it here.

Answer 6

I get the <= part, namely
if f(x)=x, then g-1(f(g(x)))=g-1(x(g(x)))=g-1(g(x))=x

The => part is not so obvious to me.
I only get, after all the substitutions, and raising to the third power, x^3+2 = (x+2)^3, for which the solution is x=0 or x=-1.

Answer 7

i am confused right away because it is not true that
\[f\circ g = g\circ g\]

Answer 8

matter of fact,the more i look at it the less sense it makes. first they define a specific f, namely
\[f(x)=x^2+2x\] and then they seem to use f to mean something completely different, the identity function

Answer 9

i think this might be an answer to a different question, because nothing seems to follow from what is stated.

\[f(x)=x^2+2x,g(x)=x^3,f\circ g(x)=f(x^3)=x^5+2x^3\] whereas
\[g\circ f(x)=g(x^2+2x)=(x^2+2x)^3=x^6+6 x^5+12 x^4+8 x^3\]

Answer 10

we solve for x by subtracting x from both sides????
don't give this a second thought, it is gibberish

Answer 11

it is easy enough to find the roots of both of these.
\[x^5+2x^2=x^3(x^2+2)=0\] solution is x= 0 since
\[x^2+2>2\] for all x

Answer 12

\[(x^2+2x)^3=0\implies x^2+2x=0\implies x =0 \text{ or } x=-1\]

Answer 13

yeah umm idk but answer got upvoted 7 times. I will try get the link of the problem.


http://math.stackexchange.com/questions/101586/let-fx-x22x-and-gx-x3-finding-the-roots-of-f-circ-gx-g-circ-fx
'Second answer'

Answer 14

yes i see the answer, the third one from alex becker.
it is crap, ignore it

Answer 15

I can't see alex becker's answer ?!

Answer 16

He deleted it probably.

Answer 17

But I have posted it on the first post in this thread. You can read it.

Answer 18

Aha, he probably realized.