## Agl202 one year ago 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.

1. SolomonZelman

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

2. 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))....

3. SolomonZelman

this migh be confusing, say so if it is.

4. Agl202

What do I do next?

5. SolomonZelman

evaluate each composition

6. Agl202

how do i do that?

7. SolomonZelman

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

8. SolomonZelman

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

9. Agl202

how do I find x?

10. SolomonZelman

you don't need to

11. SolomonZelman

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

12. SolomonZelman

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

13. Agl202

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

14. Agl202

So, D. is the correct choice?

15. 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?

16. 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.)

17. Agl202

no I didn't.

18. SolomonZelman

how did you get 19 and 32 then?

19. Agl202

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

20. SolomonZelman

I expanded, because: |dw:1441674897461:dw|

21. SolomonZelman

can you see the picture fully?

22. Agl202

Oh... K makes sense.

23. SolomonZelman

yes, and same technique I applied when expanded the parenthesis in: g(f(x))=6(2x+3)+2

24. 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) ?

25. Agl202

Yes, makes sense.

26. SolomonZelman

Ok, and as you can notice, g(f(x)) is a shift 13 units up from f(g(x)). Right?

27. Agl202

yes

28. SolomonZelman

And thus, g(f(x)) is going to be 13 {units} greater than f(g(x)), FOR ANY VALUE OF x.

29. SolomonZelman

And this is why choice D is right.

30. 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

31. SolomonZelman

Oh, you will get it:)