## anonymous one year ago For the functions f(x) = 2x − 6 and g(x) = 5x + 1, which composition produces the greatest output? Both compositions produce the same output. Neither composition produces an output. f(g(x)) produces the greatest output. g(f(x)) produces the greatest output. @mertsj @jim_thompson5910

1. jim_thompson5910

are you able to determine f(g(x)) ?

2. anonymous

I have trouble with it.

3. UsukiDoll

maybe we can write it differently like instead of f(g(x)) we can write f o g

4. UsukiDoll

and for g(f(x)) -> g o f

5. UsukiDoll

I had to read these questions from right to left when I first did them

6. jim_thompson5910

to find f(g(x)), you first replace every x with g(x) $\Large f(x) = 2x - 6$ $\Large f({\color{red}{g(x)}}) = 2({\color{red}{g(x)}}) - 6$ then on the right side, you replace g(x) with what its definition is. In this case, g(x) = 5x+1 $\Large f({\color{red}{g(x)}}) = 2({\color{red}{5x+1}}) - 6$ making sense?

7. anonymous

Kinda, yes

8. jim_thompson5910

what does 2(5x+1) - 6 simplify to?

9. UsukiDoll

or how I like to put it ... insert your entire g(x) inside the x of the f(x) $f(x) = 2x-6$ $g(x) = 5x+1$ $2(5x+1)-6$ then use distribution

10. anonymous

Distribute the 2 in the parenthesis?

11. jim_thompson5910

yes, then what?

12. UsukiDoll

distribute the 2 for (5x+1) and then combine like terms

13. anonymous

Is it possible to multiply 5x by 2?

14. jim_thompson5910

yes you will multiply the outer 2 by each term inside

15. UsukiDoll

distribution is multiplication so multiply 2 times 5x and 2 x 1

16. jim_thompson5910

|dw:1434845209689:dw|

17. anonymous

(10x + 2)-6

18. jim_thompson5910

you can toss those parenthesis since they aren't needed

19. UsukiDoll

correct 10x+2-6 so now we need 2-6

20. anonymous

-4

21. UsukiDoll

$2(5x+1)-6 \rightarrow 10x+2-6 \rightarrow 10x-4$

22. UsukiDoll

yes so that's our f(g(x)) or f o g 10x - 4

23. anonymous

How do I translate that to the answer choices.

24. UsukiDoll

now we need g(f(x)) or g o f it means place your entire f(x) equation inside the x for g(x) $f(x) = 2x-6$ $g(x) =5x+1$

25. jim_thompson5910

are you able to use these steps to figure out g(f(x)) ?

26. anonymous

can't wait to see the answer to ", which composition produces the greatest output?"

27. UsukiDoll

maybe after we have the new equation we let x = any number and whoever has the highest number has the highest output idk just guessing

28. anonymous

this is like the seventh time i have seen this question written by a mathematical illiterate what does "produces the greatest output" mean? it means nothing

29. UsukiDoll

blame the publishers.

30. anonymous

I'm really confused.

31. anonymous

you ought to be !

32. anonymous

How do I set up g(f(x))?

33. UsukiDoll

anyway... we need g(f(x)) or g o f which means we place our entire f(x) function into the x's of the g(x) function

34. UsukiDoll

recall that our functions are f(x) = 2x-6 and g(x) = 5x+1

35. anonymous

5(2x-6)+1 ?

36. jim_thompson5910

Start with g(x) and replace EVERY x with f(x) $\Large g(x) = 5x + 1$ $\Large g({\color{red}{f(x)}}) = 5({\color{red}{f(x)}}) + 1$

37. jim_thompson5910

then on the right side, replace f(x) with 2x-6 so yeah you have it, 5(2x-6) + 1

38. UsukiDoll

O_O unheard that's right... now we need to distribute and simplify

39. jim_thompson5910

simplify 5(2x-6) + 1

40. UsukiDoll

what is 5 times 2x and what is 5 x -6 ?

41. anonymous

10x+30+1

42. UsukiDoll

you have a sign error

43. UsukiDoll

5 multiplied by -6 = ?

44. anonymous

Oops, yeah, I dont have my glasses on. One sec.

45. UsukiDoll

xD it's ok.. I had trouble reading my math book last semester.. careless mistakes on my midterms.. xD

46. anonymous

-30

47. UsukiDoll

yes $10x-30+1$

48. UsukiDoll

so now what is -30+1 ?

49. anonymous

-29

50. UsukiDoll

mhm $10x-29$ is our g(f(x)) and $10x-4$ is our f(g(x))

51. anonymous

So how does that translate to my answer choices.

52. anonymous

Which is more?

53. jim_thompson5910

Let's say x = 0. What is f(g(x)) equal to? What is g(f(x)) equal to?

54. anonymous

10x-4 10x+-29

55. jim_thompson5910

in each case, replace x with 0 and compute

56. UsukiDoll

subsitute x = 0 into both equations

57. anonymous

0-4 0+-29

58. UsukiDoll

ok so what's 0-4 and 0-29?

59. anonymous

4 29

60. UsukiDoll

uhhhhh... no... that's positive numbers... we are dealing with negative numbers

61. UsukiDoll

think of it as ... you are charged \$4 but don't have any money ... you have a negative balance

62. anonymous

-4 -29

63. UsukiDoll

yes :)

64. UsukiDoll

now... bigger negative numbers are much smaller in value than single digit negative numbers

65. anonymous

so, f(g(x))!!

66. UsukiDoll

yeah... -4 is bigger than -29

67. anonymous

Thank you.