## anonymous one year ago Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = the quantity x minus seven divided by the quantity x plus three. and g(x) = quantity negative three x minus seven divided by quantity x minus one.

1. anonymous

$f(x)=\frac{ x-7 }{ x+3 } g(x)=\frac{ -3x-7 }{ x-1 }$

2. anonymous

prepare to do a raft of algebra

3. anonymous

4. anonymous

$f(x)=\frac{ x-7 }{ x+3 }$ and $g(x)=\frac{ -3x-7 }{ x-1 }$ right?

5. anonymous

yes

6. anonymous

this is amazingly easy for me to write the composition one way by using copy and paste i am going to past $$g(x)=\frac{ -3x-7 }{ x-1 }$$where every i see an $$x$$ in $$f(x)=\frac{ x-7 }{ x+3 }$$

7. anonymous

$f(x)=\frac{ x-7 }{ x+3 }$ $f(g(x))=\frac{ \frac{ -3x-7 }{ x-1 }-7 }{ \frac{ -3x-7 }{ x-1 }+3 }$

8. anonymous

okay i did that and now do i just solve?

9. anonymous

now comes the raft of algebra part

10. anonymous

which will end up with an orgy of cancellation since your answer should be $$x$$

11. anonymous

it is not really that bad multiply top and bottom of that complex fraction by $$x-1$$ to clear the denominator don't forget the distributive law

12. anonymous

okay let me just do it real quick

13. anonymous

would i do the same for the g(f(x)) = x. later on?

14. anonymous

yes, just clear the denominators then all will go bye bye

15. anonymous

ah thank you so much!

16. anonymous

yw

17. anonymous

wait im so lost what happens with the -7 and 3?

18. anonymous

don't forget the distributive law !!

19. anonymous

????

20. anonymous

$\frac{ \frac{ -3x-7 }{ x-1 }-7 }{ \frac{ -3x-7 }{ x-1 }+3 }\times \frac{x-1}{x-1}$ $=\frac{-3x-7(x-1)}{-3x-7+3(x-1)}$ is a start

21. anonymous

the distributive law in action denominators go, but you still have to distribute now distribute again, the -7 up top and the 3 below then combine like terms

22. anonymous

okay at the denominator should i add the 7 and 3 before distrubuting or no?

23. anonymous

distribute first

24. anonymous

numerator should be $$-10x+7$$ if you do it correctly

25. anonymous

okay just a min

26. anonymous

the denom i got -10

27. anonymous

yeah that is right, and i lied, the numerator is $$-3x-7-7(x-1)=-3x-7-7x+7=-10x$$

28. anonymous

and of course $$\frac{-10x}{-10}=x$$ as needed

29. anonymous

ah okay thats easy in reality thanks! ill do the other side now

30. anonymous

good luck it is real similar

31. anonymous

i just am stuck a little after muliplying x+3/x+3

32. anonymous

$g(x)=\frac{ -3x-7 }{ x-1 }$ $g(f(x))=\frac{ -3\frac{ x-7 }{ x+3 }-7 }{ \frac{ x-7 }{ x+3 }-1 }$

33. anonymous

34. anonymous

multiply by $$x-3$$ what did you get in the numerator ?

35. anonymous

why x-3?

36. anonymous

typo i meant $$x+3$$

37. anonymous

okay making sure ahha

38. anonymous

-3(x-7)-7 is my numerator?

39. anonymous

before distributing etc you should be looking at $-3(x-7)-7(x+3)$

40. anonymous

forgot that distributive law didn't you?

41. anonymous

ah yes

42. anonymous

it is the distributive LAW, not the distributive option

43. anonymous

hahaha yes so my numerator is:(-3x+21)-7x-21

44. anonymous

numerator comes to -10x

45. anonymous

got it perfect! thanks!

46. anonymous

yw