For f(x) = x^2+4 and g(x)=x^2-2, how would you find (f*g)(x), (g*f)(x), and (f*g)(4)?
https://i.imgur.com/zS0XkTh.png

- anonymous

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- UsukiDoll

for (f o g) (x) that means to plug in the g(x) function for every x you find in the f(x)
so it should look like
\[\large ( f \cdot g)(x) = (x^2-2)^2+4\]
then just expand the left part of the equation
you need this part before you can evaluate the result when x = 4
for (g o f ) (x) it means plug in your f(x) equation for all x's inside the g(x) function so we have something like this
\[( g \cdot f)(x) = (x^2+4)^2-2\]

- anonymous

So @UsukiDoll, after writing out the first one you can just substitute 4 for X and get something like this?
(4^2-2)^2 +4
which is
(16-2)^2 +4
which becomes
(14^2) +4
then then so on,
Which gives me 200, so 200 would be the answer for C?

- UsukiDoll

Your question is asking for parts.. like for part a evaluate when we have (f o g )(x) and then we need that result to answer part c which is evaluate when x=4..

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- anonymous

Right, which is why I substituted 4 as X into the answer to the first one.

- UsukiDoll

ah. I see.. so
we can just leave this unexpanded
when x = 4

- UsukiDoll

\[\large ( f \cdot g)(4) = (4^2-2)^2+4\]
\[\large ( f \cdot g)(4) = (16-2)^2+4\]
\[\large ( f \cdot g)(4) = (14)^2+4\]
\[\large ( f \cdot g)(4) = 196+4=200\]
I got 200 too.

- anonymous

So a is (f⋅g)(x)=(x2−2)2+4,
b is (g⋅f)(x)=(x2+4)2−2,
and the last is 200.
Thanks, this was waaay lass complicated then I thought it would be. I think I got it, lemme know if I'm missing something!

- UsukiDoll

I think we need to expand a bit for a and b...

- UsukiDoll

\[( g \cdot f)(x) = (x^2+4)(x^2+4)-2\]
\[\large ( f \cdot g)(x) = (x^2-2)(x^2-2)+4\]

- UsukiDoll

do you know FOIL?

- UsukiDoll

the first outer inner last.

- anonymous

I'm supposed to simplify final answers, but I guess those aren't final. I know how to foil and distribute, yes.

- UsukiDoll

\[( g \cdot f)(x) = (x^2)(x^2)+4(x^2)+4(x^2)+4(4)-2\] <-

- anonymous

(x2−2)(x2−2)+4 can simplify to (x^2-4) x^2+8, right?

- UsukiDoll

and for the other one
\[\large ( f \cdot g)(x) = (x^2)(x^2)+(-2x)+(-2x)+(-2)(-2)+4\]

- UsukiDoll

whoa one at a time XD

- anonymous

okay okay xD

- UsukiDoll

\[( g \cdot f)(x) = x^4+8(x^2)+16-2\]
when you distributed/used foil/ and simplified you got up until this step right?

- UsukiDoll

\[( g \cdot f)(x) = x^4+8(x^2)+14\]

- UsukiDoll

did you get this result for part b when you did foil?

- anonymous

For part b I got to this, so yes!
(g . f)(x) = x^4+8(x^2)+14

- UsukiDoll

alright so let's get part a
\[\large ( f \cdot g)(x) = (x^2)(x^2)+(-2x)+(-2x)+(-2)(-2)+4 \]

- UsukiDoll

\[\large ( f \cdot g)(x) = x^4-4x^2+4+4\]
made a mistake on my latex. CAREFUL!

- anonymous

Simplifying that gives me 2x^2 + -2x^2 -8.

- anonymous

wait oops

- UsukiDoll

-_- do that again.

- UsukiDoll

\[\large ( f \cdot g)(x) = (x^2)(x^2)+(-2x^2)+(-2x^2)+(-2)(-2)+4\]

- UsukiDoll

\[\large ( f \cdot g)(x) = x^4-4x^2+4+4 \]

- UsukiDoll

\[\large ( f \cdot g)(x) = x^4-4x^2+8\]

- anonymous

Right, I missed the positive 8 and didn't distribute one of the 2 I think.
So x^4−4x^2+8 is A.

- UsukiDoll

yeah

- anonymous

Wait a minute, for part b if you substitute 4 now into x^4+8(x^2)+14 I get 398 instead of 200.

- UsukiDoll

umm for part c you need the result from part a

- anonymous

I'm an idiot. I get 200.

- UsukiDoll

a and c are related to each other .... so we obtain the result from part a to answer part c which was 200

- anonymous

I got confused for a second and got mixed up. Oop.

- anonymous

Thanks for the help, I think I got it now. :D

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