Quick Help:

For the functions f(x) = 2x + 3 and g(x) = 6x + 2, which composition produces the greatest output?

Neither composition produces an output.
Both compositions produce the same output.
f(g(x)) produces the greatest output.
g(f(x)) produces the greatest output.

Question

Quick Help:


For the functions f(x) = 2x + 3 and g(x) = 6x + 2, which composition produces the greatest output? 

 	
	Neither composition produces an output.
	Both compositions produce the same output.
	f(g(x)) produces the greatest output.
	g(f(x)) produces the greatest output.

solomonzelman · Answer

$\large\color{blue}{ \displaystyle f(x)=2x+3  }$
$\large\color{red}{ \displaystyle g(x)=6x+2  }$

$\large\color{blue}{ \displaystyle f(\color{red}{g(x)})=2\left( \color{red}{6x+2}\right)+3 =? }$\

$\large\color{red}{ \displaystyle g(\color{blue}{f(x)})=6\left( \color{blue}{2x+3}\right)+2 =? }$

solomonzelman · Answer

I coloered each function in its color, and here belower f(x) and g(x),
I am showing how to set up the f(g(x)) and g(f(x))....

solomonzelman · Answer

this migh be confusing, say so if it is.

agl202 · Answer

What do I do next?

solomonzelman · Answer

evaluate each composition

agl202 · Answer

how do i do that?

solomonzelman · Answer

expand the parenthesis in:

$\large\color{blue}{ \displaystyle f(\color{red}{g(x)})=2\left( \color{red}{6x+2}\right)+3  }$

solomonzelman · Answer

then simplify as much as you can (by adding like terms)

agl202 · Answer

how do I find x?

solomonzelman · Answer

you don't need to

solomonzelman · Answer

just simplify the f(g(x)) and g(f(x)) as much as you can

solomonzelman · Answer

Again, posting them so that you won't need to scrol way back

$\large\color{blue}{ \displaystyle f(\color{red}{g(x)})=2\left( \color{red}{6x+2}\right)+3 =? }$\

$\large\color{red}{ \displaystyle g(\color{blue}{f(x)})=6\left( \color{blue}{2x+3}\right)+2 =? }$

agl202 · Answer

f(g(x))= 19
g(f(x))= 32
right?

agl202 · Answer

So, D. is the correct choice?

solomonzelman · Answer

f(g(x)) = 2(6x+2)+3 = 12x+4+3 = 12x + 7
--------------------------------------
g(f(x)) = 6(2x+3)+2=12x+18+2=12x+20

did you plug in 1 for x after that?

solomonzelman · Answer

But you didn't have that x=1 for that, did you?

(Not that it matters..... g(f(x)) is shifted 13 units up from f(g(x)), and for any x it is thus g(f(x)) is greater by 13 units.)

agl202 · Answer

no I didn't.

solomonzelman · Answer

how did you get 19 and 32 then?

agl202 · Answer

I dunno, how did u get the 4 this equation? 

--> f(g(x)) = 2(6x+2)+3 = 12x+4+3 = 12x + 7

solomonzelman · Answer

I expanded, because:
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