Question

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.

Answer 1

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

Answer 2

prepare to do a raft of algebra

Answer 3

ah okay thank you for your help in advance btw :)

Answer 4

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

Answer 5

yes

Answer 6

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

Answer 7

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

Answer 8

okay i did that and now do i just solve?

Answer 9

now comes the raft of algebra part

Answer 10

which will end up with an orgy of cancellation since your answer should be \(x\)

Answer 11

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

Answer 12

okay let me just do it real quick

Answer 13

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

Answer 14

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

Answer 15

ah thank you so much!

Answer 16

yw

Answer 17

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

Answer 18

don't forget the distributive law !!

Answer 19

????

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

distribute first

Answer 24

numerator should be \(-10x+7\) if you do it correctly

Answer 25

okay just a min

Answer 26

the denom i got -10

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

good luck
it is real similar

Answer 31

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

Answer 32

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

Answer 33

yeah i did that already

Answer 34

multiply by \(x-3\) what did you get in the numerator ?

Answer 35

why x-3?

Answer 36

typo i meant \(x+3\)

Answer 37

okay making sure ahha

Answer 38

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

Answer 39

before distributing etc you should be looking at
\[-3(x-7)-7(x+3)\]

Answer 40

forgot that distributive law didn't you?

Answer 41

ah yes

Answer 42

it is the distributive LAW, not the distributive option

Answer 43

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

Answer 44

numerator comes to -10x

Answer 45

got it perfect! thanks!

Answer 46

yw