Let f[x]=(x^2)+1. Find a function g so that

[a] [fg][x]=(x^4)-1

[b] [f+g][x]=3(x^2)

[c] [f/g][x]=1

[d] f[g[x]]=9(x^4)+1

[e] g[f[x]]=9(x^4)+1

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- Oct 13th 2007, 08:08 PMGodfatherFind a Function G and F[x]
Let f[x]=(x^2)+1. Find a function g so that

[a] [fg][x]=(x^4)-1

[b] [f+g][x]=3(x^2)

[c] [f/g][x]=1

[d] f[g[x]]=9(x^4)+1

[e] g[f[x]]=9(x^4)+1 - Oct 13th 2007, 08:30 PMJhevon
**Hint:**note that if we factored this, it would be the difference of two squares, with f being one factor

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[b] [f+g][x]=3(x^2)

simply solve for $\displaystyle g(x)$

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[c] [f/g][x]=1

**Hint:**any number divided by itself is 1

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[d] f[g[x]]=9(x^4)+1

solve for $\displaystyle g(x)$

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[e] g[f[x]]=9(x^4)+1

$\displaystyle g(x) = (3x - 3)^2 + 1$ - Oct 13th 2007, 08:36 PMGodfather
**Hint:**any number divided by itself is 1

we want $\displaystyle (g(x))^2 + 1 = 9x^4 + 1$

solve for $\displaystyle g(x)$

So it would be X^2 +1 but what is x?

i can't really think of a nice hint to give you here...

$\displaystyle g(x) = (3x - 3)^2 + 1$[/quote]

How did u get that?

What are all of the x's after [f/g]**[x]** - Oct 13th 2007, 08:51 PMJhevon
your post makes no sense. separate my quotes from your questions to make sure i don't miss anything.

you said nothing after you quoted this, so i guess you got the answer

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(to the f(g(x)) question: So it would be X^2 +1?

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but what is x?

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How did u get that?

we started with $\displaystyle g(x) = 9x^4 + 1$

now, $\displaystyle 9x^4 + 1 = \left( 3x^2 \right)^2 + 1$

$\displaystyle = \left[ 3 \left( x^2 {\color{blue} + 1 - 1} \right) \right]^2 + 1$ ...........note that i added and subtracted one from the x^2 factor, this is like adding zero, so i'm not changing anything.

$\displaystyle = \left[ 3 { \color{red} \left( x^2 + 1 \right) } - 3 \right]^2 + 1$ ...now do you see why i added and subtracted 1? when i multiplied out the -1, i am left with f(x) is the brackets. this is what i was after

$\displaystyle = \left[ 3 { \color{red} \left( f(x) \right) } - 3 \right]^2 + 1$

so now, i just let g(x) be the function that has f(x) plugged into it

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What are all of the x's after [f/g]**[x]**