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

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

Answer 2

bunch of algebra follows
multiply top and bottom by \(x-1\) to clear the fraction
then a raft of cancellation will ensue, ending only with \(x\)

Answer 3

can you show me its very confusing

Answer 4

That's the hard way to do it. And geez, @satellite you're a genius at typing so fast.

Answer 5

copy and past is all i did

Answer 6

and if you have an easier way i am all eyes

Answer 7

*paste

Answer 8

You simply reduce one of the expressions to a mixed number. That way it only has one x value. If you wanted to, you could reduce both of them that way.

For example:

\[\frac{x - 7}{x + 3} = \frac{x + 3 - 10}{x + 3} = \frac{x + 3}{x + 3} - \frac{10}{x + 3} = 1 - \frac{10}{x + 3}\]

Answer 9

Now you have one x to insert g(x) into f(x)

Answer 10

\[f(g(x)) = 1 - \frac{10}{\frac{-3x - 7}{x -1} + 3}\]

Answer 11

i can't wait to see this ...

Answer 12

It's easier to do on paper than type it out. I've already done this before. I helped another student with the exact same kind of problem.

Answer 13

But I at least was able to help reduce it so that you're inserting g(x) only once.

Answer 14

ok :)

Answer 15

there is no such thing as a free lunch

Answer 16

ill try to solve it

Answer 17

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

Answer 18

distribute?

Answer 19

is there a typo here?

Answer 20

oh no it is right
yeah distribute
combine like terms, etc
you will get it
but i do like @hero method

Answer 21

Basically, I re-wrote it like this:

\[1 - 10 \div \left(\frac{-3x - 7}{x - 1} + 3\right)\]\[1 - 10 \div \left(\frac{-3x - 7 + 3(x -1)}{x-1}\right)\]\[1 - 10 \times \left(\frac{x - 1}{-10}\right)\]\[1 + x - 1\]

Answer 22

but you are supposed to get \(x\)

Answer 23

\[1 - 1 + x = x\]

Answer 24

how do you get the 10

Answer 25

i am just funnin

Answer 26

the 10 was a trick

Answer 27

@melchar, look at my second post.

Answer 28

thank you!