Question

Find f(x) and g(x) so that the function can be described as y = f(g(x)).

Answer 1

\[y= \frac{ 9 }{ x^2 }+ 2\]

Answer 2

@Directrix @phi @zepdrix

Answer 3

First off, what do you think \(f(g(x))\) means?

Answer 4

putting the result of g into f

Answer 5

Yes, though it's important to note that it doesn't have to only be the result. So, let's give an example:
Say \(f(x)=\frac{1}{2}x\) and \(g(x)=\sqrt{x}\).

What is \(f(g(x))\)?

Answer 6

y=1/2 + sqrt x ?

Answer 7

@LolWolf

Answer 8

Not quite. Try again.

Here's a little better way to think about it:
\[
f(g(x))=\frac{1}{2}g(x)
\]

Answer 9

1/2 sqrt x

Answer 10

in my question, I'm pretty sure that f(x) = 9/x^2
and
g(x) = 2

Therefore, f(g(x)) = (9/x^2) + 2

Answer 11

@LolWolf

Answer 12

That is correct (on the first post). The second, you might want to change a bit.

What you can do is, if you have, say \(f(g(x))\), whenever there is an \(x\) within the function \(f\), you can plug in \(g(x)\). Work out some examples and then try to go backwards.

Say, how would you 'decompose' \(x^2+1\)?

Answer 13

f(x)= x^2 , g(x)=1

Answer 14

so what exactly is my f(x) and g(x) in my equation ?

Answer 15

@LolWolf

Answer 16

Not quite, here, one decomposition could be:
\[
f(g(x))=x^2+1=g(x)+1
\]and
\[
g(x)=x^2
\]

Now, say we *did* have:
\[
f(x)=x^2
\]and
\[
g(x)=1
\]We would then result in:
\[
f(g(x))=g(x)^2=1^2=1\ne x^2+1
\]Which is not what we intended.

Answer 17

ok, i sort of understand what you are saying.