Question

Please help! I don't understand how to do this:

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

f(g(x)) means start with f(x)= (x-7)/(x+3)
everywhere you see an x, replace it with g(x). It gets messy:

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

now simplify. One good idea is to start by multiplying the top and bottom by (x-1)

Answer 2

so if i multiply the top and bottom by x-1 would i multiply the 7 and 3 by it as well?

Answer 3

yes

Answer 4

the advantage to it, is you get rid of the denominators atop and below

Answer 5

i got -4x/10
i dont see how that proves it though

Answer 6

you goofed somewheres.... after multiplying the top by x-1 what did you get for the new top?

Answer 7

you are working with this, right ?
\[ f(g(x))= \frac{ \frac{-3x-7}{(x-1)} - 7}{ \frac{-3x-7}{(x-1)}+3} \]

Answer 8

one sec

Answer 9

yes i got 3x-7-7x+7 and then i simplified

Answer 10

did you this after you multiplied as phi suggested

$$
\cfrac{-3x-7-7(x-1)}{-3x-7+3(x-1)}
$$ ?

Answer 11

sorry I'm a bit lagged :/

Answer 12

did you get that I meant

Answer 13

it's -3x -7 -7x +7

Answer 14

ok now i got -x, is that right?

Answer 15

closer, what did you get for the bottom?

Answer 16

ahemm,

(g(x)) = x and g(f(x)) = x.

Answer 17

just kidding i got x. -3x-7+3x-3

Answer 18

yes -10x/(-10) = x

now do the same thing for
g(f(x))

Answer 19

would that start out like this?: ((7x-1/x+3)-3x-7)/((7x-1/x+3) + x-1
?

Answer 20

no, you replace all x's in g(x) with f(x)
g(x) = (-3x-7)/(x-1)

wherever you see an x, erase it, and squeeze in f(x)

Answer 21

got it thank you so much!