Agl202
  • Agl202
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.
Mathematics
jamiebookeater
  • jamiebookeater
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SolomonZelman
  • SolomonZelman
\(\large\color{blue}{ \displaystyle f(x)=2x+3 }\) \(\large\color{red}{ \displaystyle g(x)=6x+2 }\) \(\large\color{blue}{ \displaystyle f(\color{red}{g(x)})=2\left( \color{red}{6x+2}\right)+3 =? }\)\ \(\large\color{red}{ \displaystyle g(\color{blue}{f(x)})=6\left( \color{blue}{2x+3}\right)+2 =? }\)
SolomonZelman
  • SolomonZelman
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
  • SolomonZelman
this migh be confusing, say so if it is.

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Agl202
  • Agl202
What do I do next?
SolomonZelman
  • SolomonZelman
evaluate each composition
Agl202
  • Agl202
how do i do that?
SolomonZelman
  • SolomonZelman
expand the parenthesis in: \(\large\color{blue}{ \displaystyle f(\color{red}{g(x)})=2\left( \color{red}{6x+2}\right)+3 }\)
SolomonZelman
  • SolomonZelman
then simplify as much as you can (by adding like terms)
Agl202
  • Agl202
how do I find x?
SolomonZelman
  • SolomonZelman
you don't need to
SolomonZelman
  • SolomonZelman
just simplify the f(g(x)) and g(f(x)) as much as you can
SolomonZelman
  • SolomonZelman
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
  • Agl202
f(g(x))= 19 g(f(x))= 32 right?
Agl202
  • Agl202
So, D. is the correct choice?
SolomonZelman
  • SolomonZelman
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
  • SolomonZelman
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
  • Agl202
no I didn't.
SolomonZelman
  • SolomonZelman
how did you get 19 and 32 then?
Agl202
  • Agl202
I dunno, how did u get the 4 this equation? --> f(g(x)) = 2(6x+2)+3 = 12x+4+3 = 12x + 7
SolomonZelman
  • SolomonZelman
I expanded, because: |dw:1441674897461:dw|
SolomonZelman
  • SolomonZelman
can you see the picture fully?
Agl202
  • Agl202
Oh... K makes sense.
SolomonZelman
  • SolomonZelman
yes, and same technique I applied when expanded the parenthesis in: `g(f(x))=6(2x+3)+2`
SolomonZelman
  • SolomonZelman
So see how I got: `f(g(x)) = 2(6x+2)+3 = 12x+4+3 = 12x + 7` `g(f(x)) = 6(2x+3)+2=12x+18+2=12x+20` Yes (understand) / No (don't uderstand) ?
Agl202
  • Agl202
Yes, makes sense.
SolomonZelman
  • SolomonZelman
Ok, and as you can notice, g(f(x)) is a shift 13 units up from f(g(x)). Right?
Agl202
  • Agl202
yes
SolomonZelman
  • SolomonZelman
And thus, g(f(x)) is going to be 13 {units} greater than f(g(x)), FOR ANY VALUE OF x.
SolomonZelman
  • SolomonZelman
And this is why choice D is right.
Agl202
  • Agl202
Oh, I now understand. Thx for the help and understanding! :D Last year geometry was great, but Algebra 2 is a little challenge this year
SolomonZelman
  • SolomonZelman
Oh, you will get it:)

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