## anonymous one year ago f(x)= x-7/x+2 g(x)=-2x-7/x-3 find f(g(x)) and g(f(x))

1. anonymous

G(x)= -2x-7/x-1 (correction)

2. anonymous

prepare to do a raft of algebra

3. anonymous

Ok

4. anonymous

$f(x)=\frac{x-7}{x+2}\\ g(x)=\frac{-2x-7}{x-1}$

5. anonymous

that right?

6. anonymous

Yes :)

7. anonymous

ok lets to $f(g(x))$ first

8. anonymous

what i am going to do is use cut and paste to put $\frac{-2x-7}{x-1}$ everywhere i see an $$x$$ in $\frac{x-7}{x+2}$

9. anonymous

$\large f(g(x))=f(\frac{-2x-7}{x-1})=\frac{\frac{-2x-7}{x-1}-7}{\frac{-2x-7}{x-1}+2}$

10. anonymous

clear what i did? that is the first step, bunch of algebra comes next

11. anonymous

Yes I understand so far :)

12. anonymous

ok now before we do the algebra, i want to let you know that a miracle will occur the reason you were given this problem is that $$f$$ is the inverse of $$g$$ and vice versa that means when we get done we will find $f(g(x))=x$

13. anonymous

Ok :D

14. anonymous

staring here $\frac{\frac{-2x-7}{x-1}-7}{\frac{-2x-7}{x-1}+2}$ get rid of the compound fraction by multiplying top and bottom by $$x-1$$ (carefully)

15. anonymous

$\frac{-2x-7-7(x-1)}{-2x-7+2(x-1)}$ is step one

16. anonymous

notice the judicious use of parentheses

17. anonymous

now remove the parentheses using the distributive property to get $\frac{-2x-7-7x+7}{-2x-7+2x-2}$

18. anonymous

combine like terms, and since they are inverses you will get an orgy of cancellation $\frac{-9x}{-9}=x$finished

19. anonymous

method for doing the second one is similar

20. anonymous

That makes sense

21. anonymous

I am a bit confused for the second one

22. anonymous

same as the first, only inside out

23. anonymous

I put..g(f(x))= -2(x-7/x+2)-7/(x-7/x+2)-1

24. anonymous

try it and see what happens it will be easy enough to know if you are right or not, because the answer will be $$x$$

25. anonymous

yeah that looks right

26. anonymous

However after this step I wasn't positive if I should multiply both by x+2 or distribute

27. anonymous

yes multiply top and bottom by $$x+2$$ like i did before, carefully, using parentheses

28. anonymous

Ok I'll try that :)

29. anonymous

I don't think I'm doing this correctly ::

30. anonymous

:/