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

So when it says f(g(x)), it wants you to take the whole function of g(x) and plug it into every x in f(x). SO it would be done like this:
\[\frac{ \frac{ -3x-7 }{ x-1 }-7 }{ \frac{ -3x-7 }{ x-1 }+3 }\]
It's quite the tricky one, but that is what it wants you to do. Do you wanna try it or go through it together?

Answer 2

Yes please! I had gotten it that far but keep getting lost while simplifying

Answer 3

You can do this by plugging the entirety of one function into the other. For f(g(x)), let's take a look:

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

We can simply this by multiplying the problem by \[\frac{ x -1 }{ x - 1}\]
This gives us \[\frac{ -3x - 7 - 7(x - 1) }{ -3x - 7 + 3(x - 1) }\]= \[\frac{ -3x - 7 - 7x + 7 }{ -3x - 7 + 3x - 3 }\]=\[\frac{ -10x }{ -10 }\]= x

Answer 4

You can solve the problem in the other direction by doing the same thing. Multiplying both sides of the fraction by a polynomial is the trick to simplifying the problem.

Answer 5

Okay thank you!! :)
That clarifies a lot