Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x)=(x-7)/(x-3) and g(x)=(-3x-7)/(x-1)

Answer 1

hi :) do you know how to find f(g(x)) ?

Answer 2

umm I kind of remember so I have to plug in g in f(x)? Or is it the opposite

Answer 3

plug in g(x) in f(x)

Answer 4

Okay so do I solve [(-3x-7)/(x-1)-7]/[((-3x-7)/(x-1)-3]?

Answer 5

yes! go ahead :)

Answer 6

:) I got 5x/3x+2

Answer 7

how did you not get that = x ??
show your work...

Answer 8

numerator : (-3x-7)-7(x-1) = -3x-7-7x +7 = -3x-7x = -10 x

Answer 9

denominator : -3x-7 +3 (x-1) = -3x-7 +3x -3 = -10
actually, the inverse must be different, check your question, there is a sign mistake somewhere.

Answer 10

Yes! it's x+3!

Answer 11

i thought so :)

Answer 12

Sorry I'm trying to work this out I'm a bit slow

Answer 13

sure, take your time, after you finish , verify with these steps,
numerator : (-3x-7)-7(x-1) = -3x-7-7x +7 = -3x-7x = -10 x
denominator : -3x-7 +3 (x-1) = -3x-7 +3x -3 = -10
num / denom = -10x/ (-10) = x

Answer 14

For the numerator I got: (-10x-8)/(x-1) ... Do you know what I'm doing wrong?

Answer 15

if you show all your steps, i'll help spot the error.

Answer 16

-10x is correct, /(x-1) is also correct, only there shouldn't be -8

Answer 17

I took the -7 and added x-1 to the top and bottom and then put the two fractions together to get: (-3x-7-7x-1)/(x-1) and then I simplified the top

Answer 18

-7(x-1) = -7x - 7(-1) = -7x +7

Answer 19

so, -3x-7-7x+7 =... ?

Answer 20

-10x! :)

Answer 21

Thank now I'll get the denominator!

Answer 22

but did u understand how ??

Answer 23

Yeah! I just accidentally kept the 7 a negative even though I was multiplying two negatives!

Answer 24

Oh I see no I didn't distribute the 7 to the -1

Answer 25

yes! that was your error....

Answer 26

For the denominator I got -10x/x-1

Answer 27

-10 x or just -10 ?

Answer 28

Oh yeah I meant -10 :) because the -3x+3x=0 so there's no x

Answer 29

so now you got f(g(x)) = x ?
i think now you can do g(f(x)) =x also :)

Answer 30

I don't really know what the question means though! lol

Answer 31

Oh f(g(x)) is just x! And okay so for this question I would just show my work to prove that they both equal x?

Answer 32

if \(f(x)=g^{-1}(x), \\then , g(f(x))=g^{-1}(g(x))=x\)
the question only asks you to verify this for given pair of functions.
and yes, you showed f(g(x))=x already.
now just show g(f(x))=x similarly.
this would confirm, f and g are inverse of each other,,,,

Answer 33

Okay I got it now! Thank you for your help and patience! haha :)

Answer 34

welcome ^_^

Answer 35

Sorry @hartnn I'm having trouble getting x! I got -10x/x+3 for the numerator and -10/x+3 for the denominator

Answer 36

just divide numerator / denominator
x+3 gets cancelled ,
-10x/ (-10) = x

Answer 37

Whoops dumb mistake :P thanks again.

Answer 38

Hey, this is such an old post but it is the same exact question that I have asked, but there is something that bothers me cause I don't quite understand and was hoping you could clarify.

numerator : (-3x-7)-7(x-1) = -3x-7-7x +7 = -3x-7x = -10 x
denominator : -3x-7 +3 (x-1) = -3x-7 +3x -3 = -10

How come the (x-1) is by the -7 when I thought it was part of the fraction in g(x) shouldn't you divide it in the numerator.
I'm only asking this cause you seem to be the only one to help with difficult questions lol and I also just want to learn it so if I happen to come across it later on.

Answer 39

I multiplied numerator and denominator by (x-1)

so that fractions get eliminated from numerator and denominator.

\(\Large \dfrac{(\dfrac{-3x-7}{x-1})-7}{(\dfrac{-3x-7}{x-1})+3} \times \dfrac{x-1}{x-1} \)

\(\Large \dfrac{(\dfrac{-3x-7}{\cancel{x-1}}) \times \cancel {(x-1)}-7(x-1)}{(\dfrac{-3x-7}{\cancel{x-1}}) \times \cancel{(x-1)}+3(x-1)} \)

\(\Large \dfrac{(-3x-7)-7(x-1)}{(-3x-7)+3(x-1)} \)

Hope this helps :)

Answer 40

@CrzyWlfMedina
ask if you still have any doubt in any step :)

Answer 41

OMG THANK YOU SOOOO MUCH :D I can't believe I missed something so simple xD. You are awesome @hartnn