Question

Help please!

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.

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

I've only gotten this far:

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

Answer 1

@shinalcantara Maybe you can help again? Thanks!

Answer 2

it's more complicated than that.
f( g(x) ) means "everywhere you see x in
f(x)= (x - 7)/(x + 3)
replace the x with g(x), like this:
f( g(x) ) = ( g(x) -7) ) / ( g(x) +3 )

but g(x) is the complicated:
g(x) = (-3x - 7)/(x - 1)

to continue, replace g(x) with its (compacted) definition (-3x - 7)/(x - 1)

you will get a big mess, but it simplifies

Answer 3

Thanks for answering! I simplified and got this: ((3x + 7)/(x + 3))((x - 7)/(x - 1))
Please tell me if it's wrong.

Answer 4

Check please? @phi :)

Answer 5

start with
\[ \frac{ g(x) -7 }{ g(x) +3 }\]

now let's just to the top, for the now:

g(x) - 7

replace g(x) with its definition (just write it in)
what do you get ?

Answer 6

Its definition?

Answer 7

I mean use g(x)= (-3x - 7)/(x - 1) to replace g(x) in
g(x) - 7

Answer 8

(-3x - 7)/(x - 1) - 7 ..like that?

Answer 9

yes. in other words
\[ \frac{(-3x - 7)}{(x - 1)} -7 \]

now you want to simplify... you have a fraction so you need a common denominator
which in this case is (x-1)
multiply the -7 by (x-1)/(x-1)
to get
\[ \frac{(-3x - 7)}{(x - 1)}+ \frac{ -7(x - 1)}{(x - 1)} \]

can you simplify that ?

Answer 10

Sure, 1 sec

Answer 11

- (10x) / (x - 1) ?

Answer 12

ok or just -10x/(x-1)

now let's do the bottom
g(x)+3

it's the same method. can you do it?

Answer 13

I can try

Answer 14

(-3x - 7)/(x - 1) + 3

Answer 15

ok so far
now use a common denominator of (x-1)

Answer 16

so then would it be (-3x - 7)/(x - 1) + 3(x - 1)/(x - 1)

Answer 17

And then to simplify..

Answer 18

yes

Answer 19

10/(x -1)

Answer 20

almost. I get -10 /(x-1)

Answer 21

Oh ok, oops

Answer 22

so now we have
\[ \frac{\frac{ -10x}{x-1}}{\frac{ -10}{x-1} }\]

you can simplify that by "flipping" the bottom fraction and multiplying
(It comes from doing this:
\[ \frac{ top}{\frac{a}{b}} \cdot \frac{\frac{b}{a}}{\frac{b}{a}} = \frac{ top\cdot \frac{b}{a}}{1}
=top\cdot \frac{b}{a}
\]

Answer 23

Ok, I see

Answer 24

you should be doing this
\[ \frac{-10x}{(x-1)}\cdot \frac{(x-1)}{-10} \]
can you simplify that ?

Answer 25

if you have the same thing in the top and in the bottom, they "cancel" (i.e. anything divided by itself is 1, and we can ignore 1 when we are multiplying)

Answer 26

That simplifies into just x

Answer 27

yes, so you did part 1
Confirm that f and g are inverses by showing that f(g(x)) = x

now it's the same exercise but now show
g(f(x)) = x

Answer 28

start with
g(x) = (-3x - 7)/(x - 1)

replace x with f(x):

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

Answer 29

now do the top:
-3 f(x) -7

replace f(x) with its definition f(x) = (x - 7)/(x + 3)

Answer 30

-3(x - 7)/(x + 3) -7 ?

Answer 31

Ok great, thanks @phi ! I think I can do this myself now