Question

Let g(x) = x^2 + 3 and find a function f that produces the given composition.

Answer 1

\[(f o g)(x)=x^{4}+6x^{2}+9\]

Answer 2

I'm a bit confused at what the above compositon means is \[x^{4}+6x^{2}+9 \] f(x)?

Answer 3

sorry I mean g(x)?

Answer 4

You have \(g(x) = x^2 + 3 \)

You want to "cook up" an f(x) such that you get:

\((fog)(x)=x^4+6x^2+9\)

Remember, \((fog)(x)\) is just \(f(g(x))\)

Answer 5

So you need to exaluate \(f(x^2+3 )\), and get the result:

\(x^4+6x^2+9\)

Any ideas what choice for f(x) will accomplish that?

Answer 6

In other words, you need to decide on a funcion f(x), such that:

\(\Large f(x^2+3 )=x^4+6x^2+9\)

Ask yourself, "what do I have to DO to \(\Large x^2+3 \) to get \(\Large x^4+6x^2+9\) ?

Answer 7

You'd have to square x^2+3 or (x^2+3)^2

Answer 8

Exactly... :) So THAT is what you want f(x) to do, to whatever is "inputted" into it.

So, f(x) = ??

Answer 9

x^2

Answer 10

Tah-dah! :) Very good.

Then:
\(\Large f(x^2+3 )=(x^2+3 )^2=x^4+6x^2+9\)

Answer 11

Wow, I didn't realize it was that easy. Thanks.