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

Answer 1

@surjithayer Could you help?

Answer 2

\[\Large\bf\sf \color{royalblue}{f(x)=\left(\frac{x-9}{x+5}\right)}, \qquad\qquad \color{orangered}{g(x)=\frac{-5x-9}{x-1}}\]

\[\Large\bf\sf \color{orangered}{g\left(\color{royalblue}{f(x)}\right)=\frac{-5\color{royalblue}{\left(\frac{x-9}{x+5}\right)}-9}{\color{royalblue}{\left(\frac{x-9}{x+5}\right)}-1}}\]

Answer 3

So for the composition, g of f of x, we take the function f(x) and stuff the entire thing into each x in our g function.

Answer 4

Then it's just a matter of simplifying.
Remember how to get common denominators and stuff?

Answer 5

I would be simplifying the fraction that has substituted 'x'? Would it be something like -9x/5x or no?

Answer 6

It should simplify down to this:
\[\Large\bf\sf\frac{-5\left(\frac{x-9}{x+5}\right)-9}{\left(\frac{x-9}{x+5}\right)-1}\qquad = \qquad x\]
But that's what you need to show :o

Answer 7

So all I would have to show is that ^?

Answer 8

So for example, in the denominator, \[\Large\bf\sf \frac{x-9}{x+5}-1\quad=\quad \frac{x-9}{x+5}-\frac{x+5}{x+5}\quad=\quad \frac{x-9-x-5}{x+5}=\frac{-14}{x+5}\]

Answer 9

No you have to show the steps how you get to that, it'll be a bit of work :(

Answer 10

So it would simplify to -14/x+5?

Answer 11

The denominator, yes.

Answer 12

Okay, cool. And then what would I be doing after that?

Answer 13

\[\Large\bf\sf\frac{-5\left(\frac{x-9}{x+5}\right)-9}{\color{orangered}{\left(\frac{x-9}{x+5}\right)-1}}\quad=\quad\frac{-5\left(\frac{x-9}{x+5}\right)-9}{\color{orangered}{\dfrac{-14}{x+5}}}\]

Answer 14

After that? :O simply the numerator.

Answer 15

Blah I gotta go get ready for school and work :d soz

Answer 16

It's okay, thank you so much!

Answer 17

You should be left with, \[\frac{ -14x }{ x + 5 }\] after simplifying the numerator.

Answer 18

See how they cancel out? :D