Let f(x) = 4x2 + x + 1 and g(x) = x2 – 2. Find g(f(x)). Show each step of your work.
Help please

- anonymous

Let f(x) = 4x2 + x + 1 and g(x) = x2 – 2. Find g(f(x)). Show each step of your work.
Help please

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- schrodinger

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- anonymous

@marihelenh can you help me with this one too?

- anonymous

@freckles

- marihelenh

You would replace what you have for f(x) into the x position of g(x).
\[\left( 4x ^{2}+x+1 \right)^{2}-2\]
Does that make sense?

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## More answers

- anonymous

but where does the x^2-2 go

- anonymous

the -2 is there but the x^2 i only see the ^2 added but not the x

- marihelenh

The f(x) becomes a function of g(x), so you just replace what f was equal to into wherever the x is in the g equation.

- anonymous

oh i think i get it

- anonymous

yes i get it :D

- marihelenh

Ok great! It is a little difficult trying to explain on here sometimes.

- anonymous

4x^4+x^3-1 would this be the final answer?

- anonymous

@marihelenh

- marihelenh

Just one sec, let me check.

- marihelenh

No, the answer is actually even longer.

- marihelenh

To find the answer you would have to do,
\[(4x ^{2}+x+1)(4x ^{2}+x+1) -2\]

- anonymous

This is so hard, i thought i understood but now i don't.

- marihelenh

Do you have to simplify it?

- anonymous

It says Find g(f(x)) so I think I have to but maybe not

- marihelenh

Well, if you don't have to, you already have your answer, but I will help you simplify it. Just hang in with me. It will seem confusing, but we will get through it and to the answer. OK?

- anonymous

ok, thank you

- marihelenh

The first thing you should do, is multiply the first term of the first set by the second set.
|dw:1436743172709:dw|

- anonymous

16x^5?

- marihelenh

It would be 16x^4 + 4x^3 + 4x^2

- marihelenh

Then the same thing with the second term
|dw:1436743368652:dw|
4x^3 + x^2 + x

- anonymous

4 times 4 = 16 then 16^4 times x = 16x^5 or the x doesnt add ^1?

- marihelenh

No, it doesn't. You add the current exponents, so it would be 2 + 2 = 4.

- anonymous

oh ok

- marihelenh

Then you would do it with the third term.
|dw:1436743540161:dw|
4x^2 + x + 1

- anonymous

and what happens to the -2?

- marihelenh

That will get added on in the end when we put all of the terms together.

- anonymous

so final answer is this 16x^4 + 4x^3 + 4x^2 - 2 ?

- marihelenh

That is how you multiply the two trinomials out, so then add them all together.
\[16x ^{4}+4x ^{3}+4x ^{2}+4x ^{3}+x ^{2}+x+4x ^{2}+x+1-2\] (I added in the -2)

- anonymous

^that last part is so confusing

- marihelenh

Add like terms
\[16x ^{4}+8x ^{3}+9x ^{2}+2x-1\]

- marihelenh

Do you understand how I got the three different terms on the pictures by multiplying them out?

- anonymous

yes that part i understand

- anonymous

The f(x) becomes a function of g(x) so you just replace what f was equal to into wherever the x is in the g equation.
Then you get: (4x^2+x+1)^2(4x^2+x+1)^2-2
then multiply 4x^2 by all terms of second equation to get 16x^4
now multiply x by all terms of second equation to get 4x^3
repeat now with last term to get 4x^2
you get 16x^4 + 4x^3 + 4x^2
now to add the -2 you add like terms to get
16x^4 + 8x^3 + 9x^2 + 2x - 1
would this be good for the answer ?

- marihelenh

I just put the nine different terms together and then added the -2.

- marihelenh

Do you just have to explain how to get the answer?

- anonymous

yes

- anonymous

and get the answer too

- marihelenh

Ok, well then let me just fix what you have, you were pretty close.

- anonymous

ok

- marihelenh

then multiply 4x^2 by all terms of second equation to get 16x^4+4x^3+4x^2
now multiply x by all terms of second equation to get 4x^3+x^2+x
repeat now with last term to get 4x^2+x+1
you get 16x^4+4x^3+4x^2+4x^3+x^2+x+4x^2+x+1
now to add the -2 and add like terms to get
16x^4+8x^3+9x^2+2x-1

- marihelenh

The first three lines were good. You just had to add a little extra into the next ones, but it was all looking good.

- anonymous

yeah i missed a couple of things,
thank you so much for your time :D

- marihelenh

It's not a problem. I am soooo sorry for how long it took and how confusing it was.

- anonymous

I don't care as long as I get it, thank you.

- marihelenh

You're welcome

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