Question

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

F(x)=x-9/x+5

g(x)=-5x-9/x-1

Answer 1

just sub f(x) into g(x) and verify that it simplifies down to x
then sub g(x) into f(x) and verify that it simplifies down to x

Answer 2

you also, know they are inverses when you graph them and their line of symmetry is y=x

Answer 3

I did that for f(x) and sub f(x) into g(x) i get that but then i get stumped I am not sure exactly how to do the math for solving the g(x) part

Answer 4

Another method to figure out whether two equations are inverses of each other is to take one of the equations, change the x's and y's from each other, and then manipulate the equation to solve for y. For example let's take the second equation g(x).

y=-5x-9/x-1
Just change the x's into y's and the y's into x's like this
x=-5y-9/y-1
Now take this equation and manipulate it back to have y on one side like this
x(y-1)=-5y-9
xy-x=-5y-9
xy+5y=x-9
y(x+5)=x-9
y=x-9/x+5

f(x)=x-9/x+5 which is the exact same equation we got using this method. So yes the two equations are in fact inverses of each other.

Answer 5

I know that but I have to make the equal x! it wouldnt be a problem if it werent for that

Answer 6

@Iamaguineapig

Answer 7

Okay well then what you do is what precal said you substitute one of the equations into the other and you should get it to just equal x.

Answer 8

it is a raft of algebra is what you need to do

Answer 9

\[F(x)=\frac{x-9}{x+5}\]
\[g(x)=\frac{-5x-9}{x-1}\] right?

Answer 10

so
\[f(g(x))=f\left(\frac{-5x-9}{x-1}\right)=\frac{\frac{-5x-9}{x-1}-9}{\frac{-5x-9}{x-1}+5}\]

Answer 11

now comes the raft of algebra part
i can walk you thought it if you like, or you can try it yourself
when you get rid of that nasty compound fraction, combine like terms etc you will get a ton of cancellation and be left only with \(x\)

Answer 12

yes I got that but my problem is with g(x) I am just struggling so much with the math with it I got the f(x) part done

Answer 13

i am not sure what you mean by "i got the f(x) part done"
this is what you need to show \[f(g(x))=f\left(\frac{-5x-9}{x-1}\right)=\frac{\frac{-5x-9}{x-1}-9}{\frac{-5x-9}{x-1}+5}=x\]
did you get that part?

Answer 14

yes I got that so I was able to solve f(g(x))=x but I am struggling with the g(f(x)) part

Answer 15

@satellite73

Answer 16

ooh i see

Answer 17

the algebra will be very similar
\[g(f(x))=g\left(\frac{x-9}{x+5}\right)=\frac{-5(\frac{x-9}{x+5})-9}{\frac{x-9}{x+5}-1}\]

Answer 18

did you get that far?

Answer 19

yes, I got that far then that is where I got stuck! @satellite73

Answer 20

not sure how you did the other one, but this one is similar
get rid of compound fraction by multiplying top and bottom by \(x+5\)

Answer 21

you should get
\[\frac{-5(x-9)-9(x+5)}{x-9-(x+5)}\]

Answer 22

I did it the same way, the math there was easier I am not sure what is stumping me here but something is!:/

Answer 23

maybe a judicious use of parentheses and the distributive law
if you did not put in the parentheses it is easy to make a mistake

Answer 24

if you got what i wrote above you are just about done
remove parentheses top and bottom you get
\[\frac{-5x+35-9x-35}{x-9-x-5}\]

Answer 25

wouldnt it be 45? lol and okay and from there I just cancel out then how do I solve past that to get x? sorry I sound stupid lol

Answer 26

@satellite73

Answer 27

yeah my arithmetic sucks
whatever it is it add to zero

Answer 28

you should end up with \(\frac{-14x}{-14}\) in the next step, which is \(x\)

Answer 29

oh okay Thank you! that actually really helps and I see where I messed up now! Thanks @satellite73

Answer 30

yw